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These inequalities can represent ranges of values rather than single points, and they are used to model situations where constraints exist, making them essential in understanding how to evaluate and compare different scenarios.","relatedTerms":[{"term":"Boundary Line","definition":"The line that represents the equality part of a linear inequality when graphed, which divides the coordinate plane into two regions.","keyTermSlug":null},{"term":"Feasible Region","definition":"The area on a graph where all constraints of a system of inequalities are satisfied, representing all possible solutions.","keyTermSlug":null},{"term":"Objective Function","definition":"A function that needs to be maximized or minimized in linear programming, usually subject to various constraints represented by linear inequalities.","keyTermSlug":"objective-function"}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e44e","slug":"solution-set","subjectSlug":"contemporary-math","term":"solution set","definition":"A solution set is the collection of all values that satisfy a given equation or inequality. In mathematical contexts, this term is crucial as it helps identify all possible answers that make an equation true or meet the conditions of an inequality, creating a clearer understanding of relationships between variables.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement indicating that two expressions are not equal, often using symbols like <, >, ≤, or ≥.","keyTermSlug":null},{"term":"Linear Equation","definition":"An equation in which the highest exponent of the variable(s) is one, typically represented in the form y = mx + b.","keyTermSlug":null},{"term":"Feasible Region","definition":"The area on a graph that represents all possible solutions to a system of inequalities, bounded by the lines defined by those inequalities.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e460","slug":"≥","subjectSlug":"contemporary-math","term":"≥","definition":"The symbol '≥' represents the concept of 'greater than or equal to' in mathematics, establishing a relationship between two values. This symbol is crucial in expressing linear inequalities, where one side of the inequality can either exceed or be equal to the other side. It helps define boundaries in mathematical expressions and is foundational in various applications like optimization and systems of inequalities.","relatedTerms":[{"term":"Linear Inequality","definition":"An inequality that involves a linear expression and can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c.","keyTermSlug":"linear-inequality"},{"term":"Boundary Line","definition":"A line that represents the solutions of an equation or inequality in a coordinate system, separating different regions where different conditions apply.","keyTermSlug":null},{"term":"Feasible Region","definition":"The set of all possible points that satisfy a given set of inequalities, representing potential solutions for optimization problems.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e46c","slug":"greater","subjectSlug":"contemporary-math","term":">","definition":"The symbol '>' represents a greater than relation in mathematics, indicating that one value is larger than another. This concept is fundamental when dealing with inequalities, allowing us to express and analyze relationships between numbers or expressions. Understanding this symbol is essential for solving linear inequalities and systems of inequalities, as well as for evaluating fairness in voting methods where comparisons are necessary.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement that compares two values or expressions, indicating that one is greater than, less than, or not equal to the other.","keyTermSlug":null},{"term":"Boundary Line","definition":"In graphing systems of inequalities, it is the line representing the equation of the inequality, which helps to define the regions where the inequality holds true.","keyTermSlug":null},{"term":"Voting Power","definition":"The effectiveness of a voter in influencing the outcome of an election, often analyzed using methods that incorporate the concept of greater than in determining fairness.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"lJYAiQbp5IrpGBA6","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e45a","slug":"≤","subjectSlug":"contemporary-math","term":"≤","definition":"The symbol '≤' represents a relationship in mathematics indicating that one value is less than or equal to another. This symbol is crucial for expressing inequalities and understanding the boundaries of solutions in various mathematical contexts, helping to determine feasible solutions, especially in optimization problems.","relatedTerms":[{"term":"<","definition":"The symbol '<' denotes that one value is strictly less than another, indicating a comparison without equality.","keyTermSlug":"less"},{"term":"Inequality","definition":"An inequality is a mathematical statement that indicates the relative size or order of two values, using symbols like '≤', '<', '≥', and '>'.","keyTermSlug":null},{"term":"Feasible Region","definition":"The feasible region is the set of all possible points that satisfy a given set of inequalities, often represented graphically in coordinate systems.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e466","slug":"less","subjectSlug":"contemporary-math","term":"<","definition":"The symbol '<' represents a mathematical inequality indicating that the value on the left is less than the value on the right. This symbol is used to compare two quantities and helps in formulating linear inequalities, which are essential in both real-world applications and graphical representations of relationships between variables.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement that shows the relationship between two expressions that are not equal, often using symbols such as <, >, ≤, or ≥.","keyTermSlug":null},{"term":"Boundary Line","definition":"A line that represents the solution set of an inequality in a graph, dividing the coordinate plane into regions where the inequality holds true.","keyTermSlug":null},{"term":"Solution Set","definition":"The set of all possible values that satisfy a given inequality or system of inequalities.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9cc4a7237c55af3e4cc","slug":"y-intercept","subjectSlug":"contemporary-math","term":"y-intercept","definition":"The y-intercept is the point where a graph intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. This key feature helps to understand linear relationships, curves, and data trends, providing crucial information for graphing and analyzing equations across various mathematical contexts.","relatedTerms":[{"term":"Slope","definition":"The slope indicates the steepness or incline of a line, calculated as the change in the y-value over the change in the x-value between two points.","keyTermSlug":null},{"term":"Linear Equation","definition":"A linear equation is an equation that forms a straight line when graphed, typically expressed in the form $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.","keyTermSlug":null},{"term":"Graph","definition":"A graph is a visual representation of data or equations, displaying relationships between variables on a coordinate plane.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"JvGZRkoJlcjKJflh","type":"content"},{"id":"RyXDp1TollCLll93","type":"content"},{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"RNSQDsk12v3iW4Bi","type":"content"}]},{"_id":"66bbc9cb4a7237c55af3e4a2","slug":"feasible-region","subjectSlug":"contemporary-math","term":"feasible region","definition":"The feasible region is the set of all possible solutions that satisfy a given set of constraints in a mathematical context. This concept is crucial when dealing with inequalities and optimization problems, as it visually represents the area where all constraints overlap. The feasible region is often graphed in two-dimensional space, showing the combinations of variables that meet all specified conditions.","relatedTerms":[{"term":"Linear Inequality","definition":"An inequality that describes a linear relationship between two variables, which can be represented graphically as a half-plane.","keyTermSlug":"linear-inequality"},{"term":"Constraint","definition":"A condition or restriction placed on the variables of an optimization problem, defining the limits within which the solution must fall.","keyTermSlug":null},{"term":"Optimal Solution","definition":"The best possible solution to an optimization problem, which maximizes or minimizes a certain objective function within the feasible region.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9d04a7237c55af3e568","slug":"quadrants","subjectSlug":"contemporary-math","term":"Quadrants","definition":"Quadrants are the four sections of a Cartesian coordinate system created by the intersection of the x-axis and y-axis. Each quadrant is designated by a number (I, II, III, IV) and contains specific combinations of positive and negative values for x and y coordinates. Understanding quadrants is essential for graphing linear equations, functions, and systems of inequalities since they determine where points lie in the coordinate plane.","relatedTerms":[{"term":"Cartesian Coordinate System","definition":"A two-dimensional number system defined by a pair of perpendicular axes (x-axis and y-axis) used to locate points in the plane.","keyTermSlug":null},{"term":"Origin","definition":"The point where the x-axis and y-axis intersect, represented as (0, 0), serving as the reference point for measuring coordinates in the Cartesian plane.","keyTermSlug":null},{"term":"Ordered Pair","definition":"A pair of numbers (x, y) used to represent a point's location in the Cartesian coordinate system, where 'x' indicates the horizontal position and 'y' indicates the vertical position.","keyTermSlug":"ordered-pair"}],"parents":[{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"RyXDp1TollCLll93","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9cb4a7237c55af3e49c","slug":"constraint","subjectSlug":"contemporary-math","term":"constraint","definition":"A constraint is a limitation or restriction that defines the boundaries within which a solution must be found. In mathematical contexts, constraints help to narrow down possible solutions by establishing conditions that must be satisfied, such as inequalities or specific values. They are crucial for solving problems, as they guide the decision-making process in various applications.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement that compares two expressions, showing that one is less than, greater than, or not equal to the other.","keyTermSlug":null},{"term":"Feasible Region","definition":"The set of all possible points that satisfy the constraints of a given problem, often visualized in a graph.","keyTermSlug":null},{"term":"Linear Programming","definition":"A method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d14a7237c55af3e58c","slug":"half-plane","subjectSlug":"contemporary-math","term":"Half-plane","definition":"A half-plane is a geometric concept that refers to one of the two regions formed when a line divides a two-dimensional plane. It is essential for understanding linear equations and inequalities, as it represents the solutions to those equations or inequalities. In particular, when dealing with linear inequalities, the half-plane can indicate all the possible values that satisfy the inequality, extending infinitely in one direction.","relatedTerms":[{"term":"Linear Equation","definition":"An equation that represents a straight line in a coordinate plane, typically in the form $$y = mx + b$$.","keyTermSlug":null},{"term":"Inequality","definition":"A mathematical statement that shows the relationship between two expressions that are not equal, using symbols like $$<$$, $$>$$, $$\\leq$$, or $$\\geq$$.","keyTermSlug":null},{"term":"Boundary Line","definition":"The line that separates the two half-planes in a two-dimensional space, which is derived from a linear equation.","keyTermSlug":null}],"parents":[{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d64a7237c55af3e67c","slug":"coordinate-plane","subjectSlug":"contemporary-math","term":"coordinate plane","definition":"A coordinate plane is a two-dimensional surface formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis), allowing for the precise representation of points using ordered pairs. Each point on the coordinate plane is identified by its coordinates, which denote its position relative to the axes, facilitating the visualization and analysis of mathematical relationships and functions.","relatedTerms":[{"term":"Ordered Pair","definition":"An ordered pair is a pair of numbers used to locate a point on a coordinate plane, typically written in the form (x, y).","keyTermSlug":"ordered-pair"},{"term":"Quadrants","definition":"The coordinate plane is divided into four quadrants, each representing a different combination of positive and negative values for x and y.","keyTermSlug":"quadrants"},{"term":"Graphing","definition":"Graphing refers to the process of plotting points, lines, or curves on the coordinate plane to visualize mathematical functions or relationships.","keyTermSlug":null}],"parents":[{"id":"RyXDp1TollCLll93","type":"content"},{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d84a7237c55af3e6ac","slug":"graphing-method","subjectSlug":"contemporary-math","term":"graphing method","definition":"The graphing method is a visual technique used to solve systems of equations or inequalities by plotting their graphs on a coordinate plane. This method allows for the identification of solutions at the points where the graphs intersect, which represent the values that satisfy all equations or inequalities in the system. It is particularly effective for illustrating the relationships between variables and understanding how changes in one variable affect another.","relatedTerms":[{"term":"Linear Equation","definition":"An equation that models a straight line when graphed, typically in the form $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.","keyTermSlug":null},{"term":"Intersection Point","definition":"The point on a graph where two lines intersect, representing a solution to a system of equations.","keyTermSlug":null},{"term":"Feasible Region","definition":"The area on a graph that satisfies all constraints of a system of inequalities, representing all possible solutions.","keyTermSlug":null}],"parents":[{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d84a7237c55af3e6ca","slug":"ordered-pair","subjectSlug":"contemporary-math","term":"Ordered Pair","definition":"An ordered pair is a pair of elements where the order of the elements matters, typically represented as (x, y) in a coordinate system. The first element, x, is known as the x-coordinate, and the second element, y, is known as the y-coordinate. This concept is crucial for representing points in a two-dimensional space, which directly connects to solving equations and inequalities involving two variables.","relatedTerms":[{"term":"Coordinate Plane","definition":"A two-dimensional surface formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis), used to plot points defined by ordered pairs.","keyTermSlug":null},{"term":"Linear Equation","definition":"An equation that models a straight line when graphed on a coordinate plane, often expressed in the form y = mx + b.","keyTermSlug":null},{"term":"Graphing","definition":"The process of representing equations or inequalities visually on a coordinate plane by plotting their corresponding ordered pairs.","keyTermSlug":null}],"parents":[{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"HQmsKjzUWGFYCXh2","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e70c","slug":"solid-boundary-line","subjectSlug":"contemporary-math","term":"solid boundary line","definition":"A solid boundary line is a type of line used in the graph of a system of linear inequalities that indicates that the points on the line are included in the solution set. This line represents the equations of the inequalities and differentiates between solutions that satisfy the inequality and those that do not. It is crucial for understanding how to graphically represent and interpret the solutions to linear inequalities in two variables.","relatedTerms":[{"term":"dashed boundary line","definition":"A dashed boundary line indicates that points on the line are not included in the solution set of an inequality.","keyTermSlug":"dashed-boundary-line"},{"term":"solution set","definition":"The solution set is the collection of all possible solutions that satisfy a given system of inequalities.","keyTermSlug":"solution-set"},{"term":"feasible region","definition":"The feasible region is the area on the graph where all the inequalities overlap, representing all possible solutions to the system.","keyTermSlug":"feasible-region"}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e712","slug":"system-linear-inequalities","subjectSlug":"contemporary-math","term":"system of linear inequalities","definition":"A system of linear inequalities is a set of two or more inequalities that involve the same variables, representing constraints on those variables. These inequalities define a region in the coordinate plane, known as the feasible region, where all solutions to the system can be found. The solutions to a system of linear inequalities can be represented graphically, and they often overlap, creating a multi-dimensional space where various conditions are satisfied simultaneously.","relatedTerms":[{"term":"Feasible Region","definition":"The area in the coordinate plane that represents all possible solutions to a system of linear inequalities.","keyTermSlug":null},{"term":"Boundary Line","definition":"The line that represents the equation of a linear inequality when converted to an equality; it separates the coordinate plane into different regions.","keyTermSlug":null},{"term":"Graphing Method","definition":"A technique used to solve a system of linear inequalities by graphing each inequality and identifying the overlapping area that satisfies all conditions.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e718","slug":"optimization-problem","subjectSlug":"contemporary-math","term":"optimization problem","definition":"An optimization problem is a mathematical situation where the goal is to find the best solution from a set of feasible solutions, often defined by constraints and objectives. In the context of linear inequalities, these problems involve maximizing or minimizing a linear objective function subject to constraints represented as inequalities. The feasible region created by these inequalities helps in identifying the optimal solution.","relatedTerms":[{"term":"Feasible Region","definition":"The set of all possible points that satisfy the given linear inequalities in an optimization problem.","keyTermSlug":null},{"term":"Objective Function","definition":"A mathematical expression that defines the quantity to be maximized or minimized in an optimization problem.","keyTermSlug":"objective-function"},{"term":"Corner Point Theorem","definition":"A principle that states the optimal solution to a linear programming problem will occur at one of the corner points of the feasible region.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e71e","slug":"linear-programming","subjectSlug":"contemporary-math","term":"linear programming","definition":"Linear programming is a mathematical technique used to optimize a linear objective function, subject to a set of linear constraints, usually in the form of inequalities. This method helps in determining the best possible outcome, such as maximizing profit or minimizing costs, while adhering to given limitations like resources or budget. By visualizing constraints as linear inequalities in a graph, solutions can be found at the vertices of the feasible region created by these inequalities.","relatedTerms":[{"term":"Objective Function","definition":"The function that defines the goal of the linear programming problem, typically representing profit or cost that needs to be maximized or minimized.","keyTermSlug":"objective-function"},{"term":"Feasible Region","definition":"The set of all possible points that satisfy the constraints of a linear programming problem, typically represented graphically as a polygon.","keyTermSlug":null},{"term":"Simplex Method","definition":"An algorithm used to solve linear programming problems by moving along the edges of the feasible region to find the optimal vertex.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"Ge5hR2kfV5NXIu80","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e724","slug":"objective-function","subjectSlug":"contemporary-math","term":"Objective Function","definition":"An objective function is a mathematical expression that defines the goal of an optimization problem, typically representing the quantity to be maximized or minimized. This function plays a central role in determining the best outcome based on constraints and resources available. By evaluating different variable values, the objective function helps identify optimal solutions within feasible regions defined by linear inequalities.","relatedTerms":[{"term":"Constraints","definition":"Constraints are the conditions or limitations that restrict the values of variables in an optimization problem, often expressed as linear inequalities.","keyTermSlug":"constraints"},{"term":"Feasible Region","definition":"The feasible region is the set of all possible points that satisfy the constraints of an optimization problem, usually represented graphically as an area on a coordinate plane.","keyTermSlug":null},{"term":"Optimal Solution","definition":"An optimal solution is the best possible outcome achieved by maximizing or minimizing the objective function while satisfying all constraints.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e72a","slug":"feasibility","subjectSlug":"contemporary-math","term":"Feasibility","definition":"Feasibility refers to the condition of being possible or practical within a given context, especially concerning the constraints outlined by linear inequalities. In scenarios involving two-variable inequalities, feasibility indicates whether a certain set of values satisfies all the conditions imposed by those inequalities. This concept is crucial for determining valid solutions in optimization problems where certain criteria must be met.","relatedTerms":[{"term":"Solution Set","definition":"The collection of all possible solutions that satisfy a given system of inequalities.","keyTermSlug":null},{"term":"Bounded Region","definition":"A closed and limited area on a graph where feasible solutions reside, typically formed by the intersection of linear inequalities.","keyTermSlug":null},{"term":"Optimal Solution","definition":"The best possible solution from the feasible set that maximizes or minimizes an objective function in linear programming.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e706","slug":"dashed-boundary-line","subjectSlug":"contemporary-math","term":"dashed boundary line","definition":"A dashed boundary line is used in graphing systems of linear inequalities to indicate that points on the line itself are not included in the solution set. This means that the line represents an inequality that is strict, such as 'greater than' or 'less than', which does not allow for equality. Understanding this concept is crucial when determining feasible regions in two-dimensional space, as it visually distinguishes between points that satisfy the inequality and those that do not.","relatedTerms":[{"term":"solid boundary line","definition":"A solid boundary line is used in graphing to indicate that points on the line are included in the solution set, representing inequalities that allow for equality, such as 'greater than or equal to' or 'less than or equal to'.","keyTermSlug":"solid-boundary-line"},{"term":"feasible region","definition":"The feasible region is the area on a graph where all constraints of a system of inequalities are satisfied, including areas defined by dashed and solid boundary lines.","keyTermSlug":"feasible-region"},{"term":"linear inequality","definition":"A linear inequality is an inequality that involves a linear expression, typically written in the form 'Ax + By < C', where A, B, and C are constants, and x and y are variables.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e700","slug":"boundary-line","subjectSlug":"contemporary-math","term":"boundary line","definition":"A boundary line is a line that separates the regions of solutions for a system of linear inequalities in two variables. It is derived from the corresponding equation of the inequality, and it plays a crucial role in determining the areas where the inequalities hold true. The boundary line can either be solid or dashed, indicating whether points on the line are included in the solution set or not.","relatedTerms":[{"term":"linear inequality","definition":"A linear inequality is an expression that shows the relationship between two variables, indicating that one variable is less than, greater than, less than or equal to, or greater than or equal to another.","keyTermSlug":null},{"term":"solution set","definition":"The solution set is the collection of all possible points (x, y) that satisfy the given linear inequalities.","keyTermSlug":"solution-set"},{"term":"feasible region","definition":"The feasible region is the area on a graph where all the constraints of a system of inequalities overlap, representing all possible solutions.","keyTermSlug":"feasible-region"}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]}]},"pageData":{"subject":{"id":"contemporary-mathematics","name":"Math for Non-Math Majors","generationMetadata":{"group":"Group 1 – OpenStax (textbooks)","level":"college undergraduate","branch":"Math","duration":"one semester","subBranch":"Math","lengthVariant":"less text","model":""}},"unit":{"id":"O0VGvlW0S1mI5HJO","publicId":"O0VGvlW0S1mI5HJO","name":"Unit 5 – Algebra","order":5,"slug":"unit-5","description":"Unit 5 - Algebra","h1":null,"active":true,"emoji":"📚","hasResources":true},"topic":{"id":"7W55pLRasXjnpRGD","name":"5.10 Systems of Linear Inequalities in Two Variables","fullNumber":"5.10"},"content":{"id":"1dYC5IVDU152NnSQ","topics":[{"id":"7W55pLRasXjnpRGD","name":"5.10 Systems of Linear Inequalities in Two Variables","fullNumber":"5.10"}],"title":"5.10 Systems of Linear Inequalities in Two Variables","desc":null,"summary":null,"type":"STUDY_GUIDE","slug":"systems-linear-inequalities-variables","date":null,"vimeoLiveLink":null,"url":null,"markdown":"Linear inequalities in two variables are powerful tools for modeling real-world constraints. They're graphed on a coordinate plane, with shaded regions representing solutions. Systems of these inequalities combine multiple constraints, creating a more complex solution set.\n\nUnderstanding how to graph and interpret these systems is crucial for solving optimization problems. By identifying variables, formulating inequalities, and visualizing the solution region, we can tackle resource allocation, production planning, and other practical challenges.\n\n## Systems of Linear Inequalities in Two Variables\n\n### Graphing linear inequality systems\n\n ###### ![fiveable_image_carousel](https://fiveable.me)\n\n- Linear inequality in two variables takes the form $ax + by < c$, $ax + by \\leq c$, $ax + by > c$, or $ax + by \\geq c$, where $a$, $b$, and $c$ are real numbers and $a$ and $b$ are not both zero\n - Inequality sign determines which half-plane is shaded\n - Dashed boundary line used for strict inequalities $<$ or $>$ ($y < 2x + 1$)\n - Solid boundary line used for inclusive inequalities $\\leq$ or $\\geq$ ($y \\leq -3x + 4$)\n- Graphing a system of linear inequalities involves:\n 1. Graph each inequality separately on the same coordinate plane\n 2. 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The symbol for intersection is ∩.","relatedTerms":[{"term":"Union","definition":"The union of two sets is a new set containing all elements that belong to either one or both of the original sets.","keyTermSlug":"union"},{"term":"Difference","definition":"The difference between two sets, A and B (denoted as A - B), contains elements that are in A but not in B.","keyTermSlug":"difference"},{"term":"Complement","definition":"The complement of a set includes all elements not in the specified set, typically within a universal set context.","keyTermSlug":"complement"}],"parents":[{"id":"yDARIVPScbciMwhQ","type":"content"},{"id":"0oieHvTeq1NY5JQn","type":"content"},{"id":"VT4pzaN2sBEAGITv","type":"content"},{"id":"X7v9o4ejMuZn8hzT","type":"content"},{"id":"LLHM6dwsBKc7hvaA","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"b3NlvctmMLrZrsHq","type":"content"},{"id":"cxfsxpGN0lCjrrbk","type":"content"}]},{"_id":"66bbc9874a7237c55af3d9fe","slug":"intersection","subjectSlug":"contemporary-math","term":"Intersection","definition":"The intersection of two or more sets is the set containing all elements that are common to each of the sets. This concept is crucial for understanding relationships between different groups, helping visualize shared traits or properties through various methods.","relatedTerms":[{"term":"Union","definition":"The union of two sets is the set containing all elements that are in either set, combining their members without duplication.","keyTermSlug":"union"},{"term":"Complement","definition":"The complement of a set consists of all elements in the universal set that are not included in the given set.","keyTermSlug":"complement"},{"term":"Subset","definition":"A subset is a set where all its elements are also contained within another set, indicating a relationship of inclusion.","keyTermSlug":"subset"}],"parents":[{"id":"yDARIVPScbciMwhQ","type":"content"},{"id":"0oieHvTeq1NY5JQn","type":"content"},{"id":"VT4pzaN2sBEAGITv","type":"content"},{"id":"X7v9o4ejMuZn8hzT","type":"content"},{"id":"LLHM6dwsBKc7hvaA","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"b3NlvctmMLrZrsHq","type":"content"},{"id":"cxfsxpGN0lCjrrbk","type":"content"}]},{"_id":"66bbc9c84a7237c55af3e430","slug":"slope","subjectSlug":"contemporary-math","term":"slope","definition":"Slope is a measure of the steepness or incline of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on that line. It plays a crucial role in understanding relationships in equations and inequalities, helping to determine whether they increase or decrease, and is essential for graphing functions and analyzing systems of equations.","relatedTerms":[{"term":"y-intercept","definition":"The point where a line crosses the y-axis, representing the value of y when x is zero.","keyTermSlug":"y-intercept"},{"term":"linear equation","definition":"An equation that describes a straight line, typically in the form y = mx + b, where m is the slope and b is the y-intercept.","keyTermSlug":"linear-equation"},{"term":"rise over run","definition":"A phrase used to describe how to calculate slope by taking the change in y (rise) and dividing it by the change in x (run).","keyTermSlug":null}],"parents":[{"id":"BYOmcBfA3ARItZ8h","type":"content"},{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"RyXDp1TollCLll93","type":"content"},{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"RNSQDsk12v3iW4Bi","type":"content"},{"id":"cxfsxpGN0lCjrrbk","type":"content"}]},{"_id":"66bbc9c84a7237c55af3e448","slug":"linear-inequality","subjectSlug":"contemporary-math","term":"Linear Inequality","definition":"A linear inequality is a mathematical expression that shows the relationship between two values where one value is not equal to the other, typically expressed in the form of $ax + b < c$, $ax + b \\leq c$, $ax + b > c$, or $ax + b \\geq c$. These inequalities can represent ranges of values rather than single points, and they are used to model situations where constraints exist, making them essential in understanding how to evaluate and compare different scenarios.","relatedTerms":[{"term":"Boundary Line","definition":"The line that represents the equality part of a linear inequality when graphed, which divides the coordinate plane into two regions.","keyTermSlug":null},{"term":"Feasible Region","definition":"The area on a graph where all constraints of a system of inequalities are satisfied, representing all possible solutions.","keyTermSlug":null},{"term":"Objective Function","definition":"A function that needs to be maximized or minimized in linear programming, usually subject to various constraints represented by linear inequalities.","keyTermSlug":"objective-function"}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e44e","slug":"solution-set","subjectSlug":"contemporary-math","term":"solution set","definition":"A solution set is the collection of all values that satisfy a given equation or inequality. In mathematical contexts, this term is crucial as it helps identify all possible answers that make an equation true or meet the conditions of an inequality, creating a clearer understanding of relationships between variables.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement indicating that two expressions are not equal, often using symbols like <, >, ≤, or ≥.","keyTermSlug":null},{"term":"Linear Equation","definition":"An equation in which the highest exponent of the variable(s) is one, typically represented in the form y = mx + b.","keyTermSlug":null},{"term":"Feasible Region","definition":"The area on a graph that represents all possible solutions to a system of inequalities, bounded by the lines defined by those inequalities.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e460","slug":"≥","subjectSlug":"contemporary-math","term":"≥","definition":"The symbol '≥' represents the concept of 'greater than or equal to' in mathematics, establishing a relationship between two values. This symbol is crucial in expressing linear inequalities, where one side of the inequality can either exceed or be equal to the other side. It helps define boundaries in mathematical expressions and is foundational in various applications like optimization and systems of inequalities.","relatedTerms":[{"term":"Linear Inequality","definition":"An inequality that involves a linear expression and can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c.","keyTermSlug":"linear-inequality"},{"term":"Boundary Line","definition":"A line that represents the solutions of an equation or inequality in a coordinate system, separating different regions where different conditions apply.","keyTermSlug":null},{"term":"Feasible Region","definition":"The set of all possible points that satisfy a given set of inequalities, representing potential solutions for optimization problems.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e46c","slug":"greater","subjectSlug":"contemporary-math","term":">","definition":"The symbol '>' represents a greater than relation in mathematics, indicating that one value is larger than another. This concept is fundamental when dealing with inequalities, allowing us to express and analyze relationships between numbers or expressions. Understanding this symbol is essential for solving linear inequalities and systems of inequalities, as well as for evaluating fairness in voting methods where comparisons are necessary.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement that compares two values or expressions, indicating that one is greater than, less than, or not equal to the other.","keyTermSlug":null},{"term":"Boundary Line","definition":"In graphing systems of inequalities, it is the line representing the equation of the inequality, which helps to define the regions where the inequality holds true.","keyTermSlug":null},{"term":"Voting Power","definition":"The effectiveness of a voter in influencing the outcome of an election, often analyzed using methods that incorporate the concept of greater than in determining fairness.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"lJYAiQbp5IrpGBA6","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e45a","slug":"≤","subjectSlug":"contemporary-math","term":"≤","definition":"The symbol '≤' represents a relationship in mathematics indicating that one value is less than or equal to another. This symbol is crucial for expressing inequalities and understanding the boundaries of solutions in various mathematical contexts, helping to determine feasible solutions, especially in optimization problems.","relatedTerms":[{"term":"<","definition":"The symbol '<' denotes that one value is strictly less than another, indicating a comparison without equality.","keyTermSlug":"less"},{"term":"Inequality","definition":"An inequality is a mathematical statement that indicates the relative size or order of two values, using symbols like '≤', '<', '≥', and '>'.","keyTermSlug":null},{"term":"Feasible Region","definition":"The feasible region is the set of all possible points that satisfy a given set of inequalities, often represented graphically in coordinate systems.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9c94a7237c55af3e466","slug":"less","subjectSlug":"contemporary-math","term":"<","definition":"The symbol '<' represents a mathematical inequality indicating that the value on the left is less than the value on the right. This symbol is used to compare two quantities and helps in formulating linear inequalities, which are essential in both real-world applications and graphical representations of relationships between variables.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement that shows the relationship between two expressions that are not equal, often using symbols such as <, >, ≤, or ≥.","keyTermSlug":null},{"term":"Boundary Line","definition":"A line that represents the solution set of an inequality in a graph, dividing the coordinate plane into regions where the inequality holds true.","keyTermSlug":null},{"term":"Solution Set","definition":"The set of all possible values that satisfy a given inequality or system of inequalities.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9cc4a7237c55af3e4cc","slug":"y-intercept","subjectSlug":"contemporary-math","term":"y-intercept","definition":"The y-intercept is the point where a graph intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. This key feature helps to understand linear relationships, curves, and data trends, providing crucial information for graphing and analyzing equations across various mathematical contexts.","relatedTerms":[{"term":"Slope","definition":"The slope indicates the steepness or incline of a line, calculated as the change in the y-value over the change in the x-value between two points.","keyTermSlug":null},{"term":"Linear Equation","definition":"A linear equation is an equation that forms a straight line when graphed, typically expressed in the form $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.","keyTermSlug":null},{"term":"Graph","definition":"A graph is a visual representation of data or equations, displaying relationships between variables on a coordinate plane.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"JvGZRkoJlcjKJflh","type":"content"},{"id":"RyXDp1TollCLll93","type":"content"},{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"RNSQDsk12v3iW4Bi","type":"content"}]},{"_id":"66bbc9cb4a7237c55af3e4a2","slug":"feasible-region","subjectSlug":"contemporary-math","term":"feasible region","definition":"The feasible region is the set of all possible solutions that satisfy a given set of constraints in a mathematical context. This concept is crucial when dealing with inequalities and optimization problems, as it visually represents the area where all constraints overlap. The feasible region is often graphed in two-dimensional space, showing the combinations of variables that meet all specified conditions.","relatedTerms":[{"term":"Linear Inequality","definition":"An inequality that describes a linear relationship between two variables, which can be represented graphically as a half-plane.","keyTermSlug":"linear-inequality"},{"term":"Constraint","definition":"A condition or restriction placed on the variables of an optimization problem, defining the limits within which the solution must fall.","keyTermSlug":null},{"term":"Optimal Solution","definition":"The best possible solution to an optimization problem, which maximizes or minimizes a certain objective function within the feasible region.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9d04a7237c55af3e568","slug":"quadrants","subjectSlug":"contemporary-math","term":"Quadrants","definition":"Quadrants are the four sections of a Cartesian coordinate system created by the intersection of the x-axis and y-axis. Each quadrant is designated by a number (I, II, III, IV) and contains specific combinations of positive and negative values for x and y coordinates. Understanding quadrants is essential for graphing linear equations, functions, and systems of inequalities since they determine where points lie in the coordinate plane.","relatedTerms":[{"term":"Cartesian Coordinate System","definition":"A two-dimensional number system defined by a pair of perpendicular axes (x-axis and y-axis) used to locate points in the plane.","keyTermSlug":null},{"term":"Origin","definition":"The point where the x-axis and y-axis intersect, represented as (0, 0), serving as the reference point for measuring coordinates in the Cartesian plane.","keyTermSlug":null},{"term":"Ordered Pair","definition":"A pair of numbers (x, y) used to represent a point's location in the Cartesian coordinate system, where 'x' indicates the horizontal position and 'y' indicates the vertical position.","keyTermSlug":"ordered-pair"}],"parents":[{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"RyXDp1TollCLll93","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9cb4a7237c55af3e49c","slug":"constraint","subjectSlug":"contemporary-math","term":"constraint","definition":"A constraint is a limitation or restriction that defines the boundaries within which a solution must be found. In mathematical contexts, constraints help to narrow down possible solutions by establishing conditions that must be satisfied, such as inequalities or specific values. They are crucial for solving problems, as they guide the decision-making process in various applications.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement that compares two expressions, showing that one is less than, greater than, or not equal to the other.","keyTermSlug":null},{"term":"Feasible Region","definition":"The set of all possible points that satisfy the constraints of a given problem, often visualized in a graph.","keyTermSlug":null},{"term":"Linear Programming","definition":"A method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d14a7237c55af3e58c","slug":"half-plane","subjectSlug":"contemporary-math","term":"Half-plane","definition":"A half-plane is a geometric concept that refers to one of the two regions formed when a line divides a two-dimensional plane. It is essential for understanding linear equations and inequalities, as it represents the solutions to those equations or inequalities. In particular, when dealing with linear inequalities, the half-plane can indicate all the possible values that satisfy the inequality, extending infinitely in one direction.","relatedTerms":[{"term":"Linear Equation","definition":"An equation that represents a straight line in a coordinate plane, typically in the form $$y = mx + b$$.","keyTermSlug":null},{"term":"Inequality","definition":"A mathematical statement that shows the relationship between two expressions that are not equal, using symbols like $$<$$, $$>$$, $$\\leq$$, or $$\\geq$$.","keyTermSlug":null},{"term":"Boundary Line","definition":"The line that separates the two half-planes in a two-dimensional space, which is derived from a linear equation.","keyTermSlug":null}],"parents":[{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d64a7237c55af3e67c","slug":"coordinate-plane","subjectSlug":"contemporary-math","term":"coordinate plane","definition":"A coordinate plane is a two-dimensional surface formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis), allowing for the precise representation of points using ordered pairs. Each point on the coordinate plane is identified by its coordinates, which denote its position relative to the axes, facilitating the visualization and analysis of mathematical relationships and functions.","relatedTerms":[{"term":"Ordered Pair","definition":"An ordered pair is a pair of numbers used to locate a point on a coordinate plane, typically written in the form (x, y).","keyTermSlug":"ordered-pair"},{"term":"Quadrants","definition":"The coordinate plane is divided into four quadrants, each representing a different combination of positive and negative values for x and y.","keyTermSlug":"quadrants"},{"term":"Graphing","definition":"Graphing refers to the process of plotting points, lines, or curves on the coordinate plane to visualize mathematical functions or relationships.","keyTermSlug":null}],"parents":[{"id":"RyXDp1TollCLll93","type":"content"},{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d84a7237c55af3e6ac","slug":"graphing-method","subjectSlug":"contemporary-math","term":"graphing method","definition":"The graphing method is a visual technique used to solve systems of equations or inequalities by plotting their graphs on a coordinate plane. This method allows for the identification of solutions at the points where the graphs intersect, which represent the values that satisfy all equations or inequalities in the system. It is particularly effective for illustrating the relationships between variables and understanding how changes in one variable affect another.","relatedTerms":[{"term":"Linear Equation","definition":"An equation that models a straight line when graphed, typically in the form $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.","keyTermSlug":null},{"term":"Intersection Point","definition":"The point on a graph where two lines intersect, representing a solution to a system of equations.","keyTermSlug":null},{"term":"Feasible Region","definition":"The area on a graph that satisfies all constraints of a system of inequalities, representing all possible solutions.","keyTermSlug":null}],"parents":[{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d84a7237c55af3e6ca","slug":"ordered-pair","subjectSlug":"contemporary-math","term":"Ordered Pair","definition":"An ordered pair is a pair of elements where the order of the elements matters, typically represented as (x, y) in a coordinate system. The first element, x, is known as the x-coordinate, and the second element, y, is known as the y-coordinate. This concept is crucial for representing points in a two-dimensional space, which directly connects to solving equations and inequalities involving two variables.","relatedTerms":[{"term":"Coordinate Plane","definition":"A two-dimensional surface formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis), used to plot points defined by ordered pairs.","keyTermSlug":null},{"term":"Linear Equation","definition":"An equation that models a straight line when graphed on a coordinate plane, often expressed in the form y = mx + b.","keyTermSlug":null},{"term":"Graphing","definition":"The process of representing equations or inequalities visually on a coordinate plane by plotting their corresponding ordered pairs.","keyTermSlug":null}],"parents":[{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"HQmsKjzUWGFYCXh2","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e70c","slug":"solid-boundary-line","subjectSlug":"contemporary-math","term":"solid boundary line","definition":"A solid boundary line is a type of line used in the graph of a system of linear inequalities that indicates that the points on the line are included in the solution set. This line represents the equations of the inequalities and differentiates between solutions that satisfy the inequality and those that do not. It is crucial for understanding how to graphically represent and interpret the solutions to linear inequalities in two variables.","relatedTerms":[{"term":"dashed boundary line","definition":"A dashed boundary line indicates that points on the line are not included in the solution set of an inequality.","keyTermSlug":"dashed-boundary-line"},{"term":"solution set","definition":"The solution set is the collection of all possible solutions that satisfy a given system of inequalities.","keyTermSlug":"solution-set"},{"term":"feasible region","definition":"The feasible region is the area on the graph where all the inequalities overlap, representing all possible solutions to the system.","keyTermSlug":"feasible-region"}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e712","slug":"system-linear-inequalities","subjectSlug":"contemporary-math","term":"system of linear inequalities","definition":"A system of linear inequalities is a set of two or more inequalities that involve the same variables, representing constraints on those variables. These inequalities define a region in the coordinate plane, known as the feasible region, where all solutions to the system can be found. The solutions to a system of linear inequalities can be represented graphically, and they often overlap, creating a multi-dimensional space where various conditions are satisfied simultaneously.","relatedTerms":[{"term":"Feasible Region","definition":"The area in the coordinate plane that represents all possible solutions to a system of linear inequalities.","keyTermSlug":null},{"term":"Boundary Line","definition":"The line that represents the equation of a linear inequality when converted to an equality; it separates the coordinate plane into different regions.","keyTermSlug":null},{"term":"Graphing Method","definition":"A technique used to solve a system of linear inequalities by graphing each inequality and identifying the overlapping area that satisfies all conditions.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e718","slug":"optimization-problem","subjectSlug":"contemporary-math","term":"optimization problem","definition":"An optimization problem is a mathematical situation where the goal is to find the best solution from a set of feasible solutions, often defined by constraints and objectives. In the context of linear inequalities, these problems involve maximizing or minimizing a linear objective function subject to constraints represented as inequalities. The feasible region created by these inequalities helps in identifying the optimal solution.","relatedTerms":[{"term":"Feasible Region","definition":"The set of all possible points that satisfy the given linear inequalities in an optimization problem.","keyTermSlug":null},{"term":"Objective Function","definition":"A mathematical expression that defines the quantity to be maximized or minimized in an optimization problem.","keyTermSlug":"objective-function"},{"term":"Corner Point Theorem","definition":"A principle that states the optimal solution to a linear programming problem will occur at one of the corner points of the feasible region.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e71e","slug":"linear-programming","subjectSlug":"contemporary-math","term":"linear programming","definition":"Linear programming is a mathematical technique used to optimize a linear objective function, subject to a set of linear constraints, usually in the form of inequalities. This method helps in determining the best possible outcome, such as maximizing profit or minimizing costs, while adhering to given limitations like resources or budget. By visualizing constraints as linear inequalities in a graph, solutions can be found at the vertices of the feasible region created by these inequalities.","relatedTerms":[{"term":"Objective Function","definition":"The function that defines the goal of the linear programming problem, typically representing profit or cost that needs to be maximized or minimized.","keyTermSlug":"objective-function"},{"term":"Feasible Region","definition":"The set of all possible points that satisfy the constraints of a linear programming problem, typically represented graphically as a polygon.","keyTermSlug":null},{"term":"Simplex Method","definition":"An algorithm used to solve linear programming problems by moving along the edges of the feasible region to find the optimal vertex.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"Ge5hR2kfV5NXIu80","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e724","slug":"objective-function","subjectSlug":"contemporary-math","term":"Objective Function","definition":"An objective function is a mathematical expression that defines the goal of an optimization problem, typically representing the quantity to be maximized or minimized. This function plays a central role in determining the best outcome based on constraints and resources available. By evaluating different variable values, the objective function helps identify optimal solutions within feasible regions defined by linear inequalities.","relatedTerms":[{"term":"Constraints","definition":"Constraints are the conditions or limitations that restrict the values of variables in an optimization problem, often expressed as linear inequalities.","keyTermSlug":"constraints"},{"term":"Feasible Region","definition":"The feasible region is the set of all possible points that satisfy the constraints of an optimization problem, usually represented graphically as an area on a coordinate plane.","keyTermSlug":null},{"term":"Optimal Solution","definition":"An optimal solution is the best possible outcome achieved by maximizing or minimizing the objective function while satisfying all constraints.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e72a","slug":"feasibility","subjectSlug":"contemporary-math","term":"Feasibility","definition":"Feasibility refers to the condition of being possible or practical within a given context, especially concerning the constraints outlined by linear inequalities. In scenarios involving two-variable inequalities, feasibility indicates whether a certain set of values satisfies all the conditions imposed by those inequalities. This concept is crucial for determining valid solutions in optimization problems where certain criteria must be met.","relatedTerms":[{"term":"Solution Set","definition":"The collection of all possible solutions that satisfy a given system of inequalities.","keyTermSlug":null},{"term":"Bounded Region","definition":"A closed and limited area on a graph where feasible solutions reside, typically formed by the intersection of linear inequalities.","keyTermSlug":null},{"term":"Optimal Solution","definition":"The best possible solution from the feasible set that maximizes or minimizes an objective function in linear programming.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e706","slug":"dashed-boundary-line","subjectSlug":"contemporary-math","term":"dashed boundary line","definition":"A dashed boundary line is used in graphing systems of linear inequalities to indicate that points on the line itself are not included in the solution set. This means that the line represents an inequality that is strict, such as 'greater than' or 'less than', which does not allow for equality. 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Topics usually include logic, probability, statistics, financial math, and geometry. You'll learn how to analyze data, make informed decisions, and solve real-world problems using math concepts. The course focuses on developing critical thinking skills and mathematical reasoning without heavy algebra or calculus.\n\n## Is Contemporary Mathematics for Non-Math Majors hard?\n\nIt's designed to be accessible for non-math majors, so it's not as brutal as calculus or advanced algebra. The concepts are more practical and relatable to real life, which makes them easier to grasp. That said, it still requires effort and engagement. Some students find the logic and problem-solving aspects challenging, but overall, it's manageable if you put in the work.\n\n## Tips for taking Contemporary Mathematics for Non-Math Majors in college\n\n1. Use [Fiveable Study Guides](https://fiveable.me/cram-mode) to help you cram 🌶️\n2. Practice, practice, practice - especially with probability and statistics problems\n3. Form study groups to tackle complex logic puzzles together\n4. Use real-life examples to understand financial math concepts (like budgeting or loans)\n5. Watch \"Moneyball\" to see how statistics are applied in sports\n6. Read \"Innumeracy\" by John Allen Paulos for a fun take on mathematical thinking\n7. Don't miss class - each topic usually builds on the previous one\n8. Use online resources like Khan Academy for extra explanations\n\n## Common pre-requisites for Contemporary Mathematics for Non-Math Majors\n\n1. College Algebra: This course covers fundamental algebraic concepts and functions. It's a foundation for many math and science courses.\n\n2. Intro to Statistics: This class introduces basic statistical methods and probability theory. It's useful for understanding data analysis and interpretation.\n\n## Classes similar to Contemporary Mathematics for Non-Math Majors\n\n1. Math for Liberal Arts: This course explores various mathematical topics and their applications in art, music, and nature. It's designed to show the beauty and relevance of math in non-technical fields.\n\n2. Quantitative Reasoning: This class focuses on developing logical thinking and problem-solving skills using math. It often includes topics like personal finance, data interpretation, and basic geometry.\n\n3. Finite Mathematics: This course covers mathematical concepts used in business and social sciences. It typically includes linear equations, matrices, linear programming, and probability theory.\n\n## Majors related to Contemporary Mathematics for Non-Math Majors\n\n1. Business Administration: Focuses on the principles of managing organizations, including finance, marketing, and operations. Students learn to make data-driven decisions and analyze market trends.\n\n2. Liberal Arts: Provides a broad education across various disciplines, including literature, history, and social sciences. Students develop critical thinking and communication skills applicable to many career paths.\n\n3. Education: Prepares students to become teachers at various levels. Includes courses on learning theories, curriculum development, and classroom management techniques.\n\n## What can you do with a degree in Contemporary Mathematics for Non-Math Majors?\n\n1. Data Analyst: Collects, processes, and performs statistical analyses of data. They help organizations make informed decisions by interpreting complex data sets and creating visual representations of findings.\n\n2. Financial Advisor: Assists clients in managing their finances, including investments, insurance, and retirement planning. They use mathematical concepts to analyze financial markets and create personalized financial strategies.\n\n3. Market Research Analyst: Studies market conditions to examine potential sales of products or services. They help companies understand what products people want, who will buy them, and at what price.\n\n## Contemporary Mathematics for Non-Math Majors FAQs\n\n1. Can this course help me in my daily life? Absolutely! You'll learn practical math skills for budgeting, understanding statistics in the news, and making informed decisions.\n\n2. Do we use calculators in this class? Usually, yes. The focus is more on understanding concepts and applying them rather than doing complex calculations by hand.\n\n3. Is there a lot of homework? It varies, but expect regular problem sets to practice the concepts. Most professors assign weekly homework to reinforce what you've learned in class.","emoji":"💯","order":null,"numResources":null,"active":true,"slug":"contemporary-math","generationMetadata":{"group":"Group 1 – OpenStax (textbooks)","level":"college undergraduate","branch":"Math","duration":"one semester","subBranch":"Math","lengthVariant":"less text","model":""}},"pageParams":{"communitySlug":"contemporary-math","unitSlug":"unit-5","contentSlug":"systems-linear-inequalities-variables","docId":"1dYC5IVDU152NnSQ"},"children":["$","$L1c",null,{"content":{"id":"1dYC5IVDU152NnSQ","topics":[{"id":"7W55pLRasXjnpRGD","name":"5.10 Systems of Linear Inequalities in Two Variables","fullNumber":"5.10"}],"title":"5.10 Systems of Linear Inequalities in Two Variables","desc":null,"summary":null,"type":"STUDY_GUIDE","slug":"systems-linear-inequalities-variables","date":null,"vimeoLiveLink":null,"url":null,"markdown":"Linear inequalities in two variables are powerful tools for modeling real-world [constraints](https://www.fiveableKeyTerm:constraints). They're graphed on a [coordinate plane](https://www.fiveableKeyTerm:coordinate_plane), with shaded regions representing solutions. Systems of these inequalities combine multiple constraints, creating a more complex [solution set](https://www.fiveableKeyTerm:solution_set).\n\nUnderstanding how to graph and interpret these systems is crucial for solving optimization problems. By identifying variables, formulating inequalities, and visualizing the solution region, we can tackle resource allocation, production planning, and other practical challenges.\n\n## Systems of Linear Inequalities in Two Variables\n\n### Graphing linear inequality systems\n\n ###### ![fiveable_image_carousel](https://fiveable.me)\n\n- [Linear inequality](https://www.fiveableKeyTerm:Linear_Inequality) in two variables takes the form $ax + by [<](https://www.fiveableKeyTerm:<) c$, $ax + by \\leq c$, $ax + by [>](https://www.fiveableKeyTerm:>) c$, or $ax + by \\geq c$, where $a$, $b$, and $c$ are real numbers and $a$ and $b$ are not both zero\n - Inequality sign determines which [half-plane](https://www.fiveableKeyTerm:Half-plane) is shaded\n - [Dashed boundary line](https://www.fiveableKeyTerm:dashed_boundary_line) used for strict inequalities $<$ or $>$ ($y < 2x + 1$)\n - [Solid boundary line](https://www.fiveableKeyTerm:solid_boundary_line) used for inclusive inequalities $\\leq$ or $\\geq$ ($y \\leq -3x + 4$)\n- Graphing a [system of linear inequalities](https://www.fiveableKeyTerm:system_of_linear_inequalities) involves:\n 1. Graph each inequality separately on the same coordinate plane\n 2. Solution set is the [intersection](https://www.fiveableKeyTerm:Intersection) of all shaded regions satisfying the inequalities\n- Solution set of a system of linear inequalities represents all ordered pairs $(x, y)$ that simultaneously satisfy every inequality in the system\n### Ordered pairs in inequality systems\n- Checking if an [ordered pair](https://www.fiveableKeyTerm:Ordered_Pair) $(x, y)$ satisfies a system of inequalities:\n - Substitute $x$ and $y$ values into each inequality in the system\n - If substitution results in true statements for all inequalities, the ordered pair is a solution\n- Ordered pair satisfying all inequalities in the system is a solution point contained within the solution set (graphically represented by the intersecting shaded regions)\n### Real-world applications of inequalities\n- Systems of linear inequalities model real-world situations involving constraints or limitations\n - Resource allocation problems (budget constraints)\n - Production limitations (manufacturing capacity)\n - Feasible regions in optimization problems (maximizing profit)\n- Modeling a real-world scenario with inequalities:\n 1. Identify variables and define them in the context of the problem\n 2. Formulate inequalities representing constraints or limitations on the variables\n 3. Graph the system of inequalities to visualize the feasible solution region\n 4. Interpret the solution set in terms of the original real-world problem\n- [Linear programming](https://www.fiveableKeyTerm:Linear_programming) problems involve optimizing an [objective function](https://www.fiveableKeyTerm:Objective_Function) subject to constraints modeled by a system of linear inequalities (maximizing profit while satisfying production constraints)\n### Key components of linear inequalities\n- Coordinate plane: The 2D plane where linear inequalities are graphed, consisting of x and y axes\n- [Quadrants](https://www.fiveableKeyTerm:Quadrants): The four regions of the coordinate plane, divided by the x and y axes\n- [Slope](https://www.fiveableKeyTerm:slope): The rate of change in a linear equation, representing the steepness of the line\n- [Y-intercept](https://www.fiveableKeyTerm:y-intercept): The point where a line crosses the y-axis, often used as a starting point when graphing\n- [Feasibility](https://www.fiveableKeyTerm:Feasibility): The property of a solution satisfying all constraints in a system of linear inequalities","cheatsheet":null,"publishDate":null,"updatedAt":"2024-06-18T06:00:00.000Z","status":"PUBLISHED","images":[{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Graphing_systems_of_linear_inequalities_in_two_variables_with_shaded_regions_and_boundary_lines%22-image014-300x297.gif","description":"Systems of Linear Inequalities | Math Modeling","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1468/2016/02/04064403/image014-300x297.gif","hostUrl":"https://courses.lumenlearning.com/albanytech-mathmodeling/chapter/read-define-solutions-to-systems-of-linear-inequalities/","altText":null,"sectionTitle":"Graphing linear inequality systems","rank":1,"height":297,"width":300,"displayWidth":150,"displayHeight":148,"contentId":"666c39a72c14cb3a37f76050","subjectId":"contemporary-mathematics"},{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Graphing_systems_of_linear_inequalities_in_two_variables_with_shaded_regions_and_boundary_lines%22-GraphingLinearInequalities2-1024x887.png","description":"3.1 Graphing Systems Of Linear Inequalites | Finite Math","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/395/2016/10/13033606/GraphingLinearInequalities2-1024x887.png","hostUrl":"https://courses.lumenlearning.com/sanjacinto-finitemath1/chapter/3-1-graphing-systems-of-linear-inequalites/","altText":null,"sectionTitle":"Graphing linear inequality systems","rank":2,"height":887,"width":1024,"displayWidth":512,"displayHeight":443,"contentId":"666c39a72c14cb3a37f76050","subjectId":"contemporary-mathematics"},{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Graphing_systems_of_linear_inequalities_in_two_variables_with_shaded_regions_and_boundary_lines%22-Screen-Shot-2016-05-13-at-2.07.01-PM-297x300.png","description":"Linear Inequalities and Systems of Linear Inequalities in Two Variables | Math Modeling","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01183208/Screen-Shot-2016-05-13-at-2.07.01-PM-297x300.png","hostUrl":"https://courses.lumenlearning.com/albanytech-mathmodeling/chapter/outcome-graphing-linear-equations/","altText":null,"sectionTitle":"Graphing linear inequality systems","rank":3,"height":300,"width":297,"displayWidth":148,"displayHeight":150,"contentId":"666c39a72c14cb3a37f76050","subjectId":"contemporary-mathematics"}],"tableOfContents":{"_id":"669e943fffd2f55460b5a097","id":"contemporary-mathematics","data":[{"id":"0mlqSYUaaNq21ONy","title":"Unit 1 - 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Understanding quadrants is essential for graphing linear equations, functions, and systems of inequalities since they determine where points lie in the coordinate plane.","relatedTerms":[{"term":"Cartesian Coordinate System","definition":"A two-dimensional number system defined by a pair of perpendicular axes (x-axis and y-axis) used to locate points in the plane.","keyTermSlug":null},{"term":"Origin","definition":"The point where the x-axis and y-axis intersect, represented as (0, 0), serving as the reference point for measuring coordinates in the Cartesian plane.","keyTermSlug":null},{"term":"Ordered Pair","definition":"A pair of numbers (x, y) used to represent a point's location in the Cartesian coordinate system, where 'x' indicates the horizontal position and 'y' indicates the vertical position.","keyTermSlug":"ordered-pair"}],"parents":[{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"RyXDp1TollCLll93","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9cb4a7237c55af3e49c","slug":"constraint","subjectSlug":"contemporary-math","term":"constraint","definition":"A constraint is a limitation or restriction that defines the boundaries within which a solution must be found. In mathematical contexts, constraints help to narrow down possible solutions by establishing conditions that must be satisfied, such as inequalities or specific values. They are crucial for solving problems, as they guide the decision-making process in various applications.","relatedTerms":[{"term":"Inequality","definition":"A mathematical statement that compares two expressions, showing that one is less than, greater than, or not equal to the other.","keyTermSlug":null},{"term":"Feasible Region","definition":"The set of all possible points that satisfy the constraints of a given problem, often visualized in a graph.","keyTermSlug":null},{"term":"Linear Programming","definition":"A method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships.","keyTermSlug":null}],"parents":[{"id":"AdSvIsUQMcQQnjlj","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d14a7237c55af3e58c","slug":"half-plane","subjectSlug":"contemporary-math","term":"Half-plane","definition":"A half-plane is a geometric concept that refers to one of the two regions formed when a line divides a two-dimensional plane. It is essential for understanding linear equations and inequalities, as it represents the solutions to those equations or inequalities. In particular, when dealing with linear inequalities, the half-plane can indicate all the possible values that satisfy the inequality, extending infinitely in one direction.","relatedTerms":[{"term":"Linear Equation","definition":"An equation that represents a straight line in a coordinate plane, typically in the form $$y = mx + b$$.","keyTermSlug":null},{"term":"Inequality","definition":"A mathematical statement that shows the relationship between two expressions that are not equal, using symbols like $$<$$, $$>$$, $$\\leq$$, or $$\\geq$$.","keyTermSlug":null},{"term":"Boundary Line","definition":"The line that separates the two half-planes in a two-dimensional space, which is derived from a linear equation.","keyTermSlug":null}],"parents":[{"id":"xHHVJNli5YMtvjMC","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d64a7237c55af3e67c","slug":"coordinate-plane","subjectSlug":"contemporary-math","term":"coordinate plane","definition":"A coordinate plane is a two-dimensional surface formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis), allowing for the precise representation of points using ordered pairs. Each point on the coordinate plane is identified by its coordinates, which denote its position relative to the axes, facilitating the visualization and analysis of mathematical relationships and functions.","relatedTerms":[{"term":"Ordered Pair","definition":"An ordered pair is a pair of numbers used to locate a point on a coordinate plane, typically written in the form (x, y).","keyTermSlug":"ordered-pair"},{"term":"Quadrants","definition":"The coordinate plane is divided into four quadrants, each representing a different combination of positive and negative values for x and y.","keyTermSlug":"quadrants"},{"term":"Graphing","definition":"Graphing refers to the process of plotting points, lines, or curves on the coordinate plane to visualize mathematical functions or relationships.","keyTermSlug":null}],"parents":[{"id":"RyXDp1TollCLll93","type":"content"},{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d84a7237c55af3e6ac","slug":"graphing-method","subjectSlug":"contemporary-math","term":"graphing method","definition":"The graphing method is a visual technique used to solve systems of equations or inequalities by plotting their graphs on a coordinate plane. This method allows for the identification of solutions at the points where the graphs intersect, which represent the values that satisfy all equations or inequalities in the system. It is particularly effective for illustrating the relationships between variables and understanding how changes in one variable affect another.","relatedTerms":[{"term":"Linear Equation","definition":"An equation that models a straight line when graphed, typically in the form $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.","keyTermSlug":null},{"term":"Intersection Point","definition":"The point on a graph where two lines intersect, representing a solution to a system of equations.","keyTermSlug":null},{"term":"Feasible Region","definition":"The area on a graph that satisfies all constraints of a system of inequalities, representing all possible solutions.","keyTermSlug":null}],"parents":[{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9d84a7237c55af3e6ca","slug":"ordered-pair","subjectSlug":"contemporary-math","term":"Ordered Pair","definition":"An ordered pair is a pair of elements where the order of the elements matters, typically represented as (x, y) in a coordinate system. The first element, x, is known as the x-coordinate, and the second element, y, is known as the y-coordinate. This concept is crucial for representing points in a two-dimensional space, which directly connects to solving equations and inequalities involving two variables.","relatedTerms":[{"term":"Coordinate Plane","definition":"A two-dimensional surface formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis), used to plot points defined by ordered pairs.","keyTermSlug":null},{"term":"Linear Equation","definition":"An equation that models a straight line when graphed on a coordinate plane, often expressed in the form y = mx + b.","keyTermSlug":null},{"term":"Graphing","definition":"The process of representing equations or inequalities visually on a coordinate plane by plotting their corresponding ordered pairs.","keyTermSlug":null}],"parents":[{"id":"9B6kZfIpnsm9SSNx","type":"content"},{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"HQmsKjzUWGFYCXh2","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e70c","slug":"solid-boundary-line","subjectSlug":"contemporary-math","term":"solid boundary line","definition":"A solid boundary line is a type of line used in the graph of a system of linear inequalities that indicates that the points on the line are included in the solution set. This line represents the equations of the inequalities and differentiates between solutions that satisfy the inequality and those that do not. It is crucial for understanding how to graphically represent and interpret the solutions to linear inequalities in two variables.","relatedTerms":[{"term":"dashed boundary line","definition":"A dashed boundary line indicates that points on the line are not included in the solution set of an inequality.","keyTermSlug":"dashed-boundary-line"},{"term":"solution set","definition":"The solution set is the collection of all possible solutions that satisfy a given system of inequalities.","keyTermSlug":"solution-set"},{"term":"feasible region","definition":"The feasible region is the area on the graph where all the inequalities overlap, representing all possible solutions to the system.","keyTermSlug":"feasible-region"}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e712","slug":"system-linear-inequalities","subjectSlug":"contemporary-math","term":"system of linear inequalities","definition":"A system of linear inequalities is a set of two or more inequalities that involve the same variables, representing constraints on those variables. These inequalities define a region in the coordinate plane, known as the feasible region, where all solutions to the system can be found. The solutions to a system of linear inequalities can be represented graphically, and they often overlap, creating a multi-dimensional space where various conditions are satisfied simultaneously.","relatedTerms":[{"term":"Feasible Region","definition":"The area in the coordinate plane that represents all possible solutions to a system of linear inequalities.","keyTermSlug":null},{"term":"Boundary Line","definition":"The line that represents the equation of a linear inequality when converted to an equality; it separates the coordinate plane into different regions.","keyTermSlug":null},{"term":"Graphing Method","definition":"A technique used to solve a system of linear inequalities by graphing each inequality and identifying the overlapping area that satisfies all conditions.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e718","slug":"optimization-problem","subjectSlug":"contemporary-math","term":"optimization problem","definition":"An optimization problem is a mathematical situation where the goal is to find the best solution from a set of feasible solutions, often defined by constraints and objectives. In the context of linear inequalities, these problems involve maximizing or minimizing a linear objective function subject to constraints represented as inequalities. The feasible region created by these inequalities helps in identifying the optimal solution.","relatedTerms":[{"term":"Feasible Region","definition":"The set of all possible points that satisfy the given linear inequalities in an optimization problem.","keyTermSlug":null},{"term":"Objective Function","definition":"A mathematical expression that defines the quantity to be maximized or minimized in an optimization problem.","keyTermSlug":"objective-function"},{"term":"Corner Point Theorem","definition":"A principle that states the optimal solution to a linear programming problem will occur at one of the corner points of the feasible region.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e71e","slug":"linear-programming","subjectSlug":"contemporary-math","term":"linear programming","definition":"Linear programming is a mathematical technique used to optimize a linear objective function, subject to a set of linear constraints, usually in the form of inequalities. This method helps in determining the best possible outcome, such as maximizing profit or minimizing costs, while adhering to given limitations like resources or budget. By visualizing constraints as linear inequalities in a graph, solutions can be found at the vertices of the feasible region created by these inequalities.","relatedTerms":[{"term":"Objective Function","definition":"The function that defines the goal of the linear programming problem, typically representing profit or cost that needs to be maximized or minimized.","keyTermSlug":"objective-function"},{"term":"Feasible Region","definition":"The set of all possible points that satisfy the constraints of a linear programming problem, typically represented graphically as a polygon.","keyTermSlug":null},{"term":"Simplex Method","definition":"An algorithm used to solve linear programming problems by moving along the edges of the feasible region to find the optimal vertex.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"Ge5hR2kfV5NXIu80","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e724","slug":"objective-function","subjectSlug":"contemporary-math","term":"Objective Function","definition":"An objective function is a mathematical expression that defines the goal of an optimization problem, typically representing the quantity to be maximized or minimized. This function plays a central role in determining the best outcome based on constraints and resources available. By evaluating different variable values, the objective function helps identify optimal solutions within feasible regions defined by linear inequalities.","relatedTerms":[{"term":"Constraints","definition":"Constraints are the conditions or limitations that restrict the values of variables in an optimization problem, often expressed as linear inequalities.","keyTermSlug":"constraints"},{"term":"Feasible Region","definition":"The feasible region is the set of all possible points that satisfy the constraints of an optimization problem, usually represented graphically as an area on a coordinate plane.","keyTermSlug":null},{"term":"Optimal Solution","definition":"An optimal solution is the best possible outcome achieved by maximizing or minimizing the objective function while satisfying all constraints.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9db4a7237c55af3e72a","slug":"feasibility","subjectSlug":"contemporary-math","term":"Feasibility","definition":"Feasibility refers to the condition of being possible or practical within a given context, especially concerning the constraints outlined by linear inequalities. In scenarios involving two-variable inequalities, feasibility indicates whether a certain set of values satisfies all the conditions imposed by those inequalities. This concept is crucial for determining valid solutions in optimization problems where certain criteria must be met.","relatedTerms":[{"term":"Solution Set","definition":"The collection of all possible solutions that satisfy a given system of inequalities.","keyTermSlug":null},{"term":"Bounded Region","definition":"A closed and limited area on a graph where feasible solutions reside, typically formed by the intersection of linear inequalities.","keyTermSlug":null},{"term":"Optimal Solution","definition":"The best possible solution from the feasible set that maximizes or minimizes an objective function in linear programming.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"},{"id":"YjcyquOjJU33Crx3","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e706","slug":"dashed-boundary-line","subjectSlug":"contemporary-math","term":"dashed boundary line","definition":"A dashed boundary line is used in graphing systems of linear inequalities to indicate that points on the line itself are not included in the solution set. This means that the line represents an inequality that is strict, such as 'greater than' or 'less than', which does not allow for equality. Understanding this concept is crucial when determining feasible regions in two-dimensional space, as it visually distinguishes between points that satisfy the inequality and those that do not.","relatedTerms":[{"term":"solid boundary line","definition":"A solid boundary line is used in graphing to indicate that points on the line are included in the solution set, representing inequalities that allow for equality, such as 'greater than or equal to' or 'less than or equal to'.","keyTermSlug":"solid-boundary-line"},{"term":"feasible region","definition":"The feasible region is the area on a graph where all constraints of a system of inequalities are satisfied, including areas defined by dashed and solid boundary lines.","keyTermSlug":"feasible-region"},{"term":"linear inequality","definition":"A linear inequality is an inequality that involves a linear expression, typically written in the form 'Ax + By < C', where A, B, and C are constants, and x and y are variables.","keyTermSlug":null}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]},{"_id":"66bbc9da4a7237c55af3e700","slug":"boundary-line","subjectSlug":"contemporary-math","term":"boundary line","definition":"A boundary line is a line that separates the regions of solutions for a system of linear inequalities in two variables. It is derived from the corresponding equation of the inequality, and it plays a crucial role in determining the areas where the inequalities hold true. The boundary line can either be solid or dashed, indicating whether points on the line are included in the solution set or not.","relatedTerms":[{"term":"linear inequality","definition":"A linear inequality is an expression that shows the relationship between two variables, indicating that one variable is less than, greater than, less than or equal to, or greater than or equal to another.","keyTermSlug":null},{"term":"solution set","definition":"The solution set is the collection of all possible points (x, y) that satisfy the given linear inequalities.","keyTermSlug":"solution-set"},{"term":"feasible region","definition":"The feasible region is the area on a graph where all the constraints of a system of inequalities overlap, representing all possible solutions.","keyTermSlug":"feasible-region"}],"parents":[{"id":"1dYC5IVDU152NnSQ","type":"content"}]}]}]}]]