3.4 Graph Linear Inequalities in Two Variables
3 min read•june 24, 2024
Linear inequalities in divide the into regions, with one region representing the . Graphing these inequalities involves drawing boundary lines and shading the appropriate based on the inequality symbol and a test point.
Understanding linear inequalities is crucial for solving real-world problems involving constraints. This knowledge forms the foundation for , where multiple inequalities create a , allowing us to optimize solutions within given limitations.
Graphing Linear Inequalities in Two Variables
Verification of two-variable inequality solutions
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- Substitute the given values for the variables into the inequality
- Inequality true after substitution point is a solution
- Inequality false after substitution point is not a solution
- Given inequality , verify if (2, 3) is a solution
- Substitute and :
- Simplify: or , which is false
- (2, 3) is not a solution to the inequality
Interpretation of linear inequality graphs
- graph divides coordinate plane into two regions (half-planes)
- One region represents points satisfying inequality (solution set)
- Other region represents points not satisfying inequality
- determined by corresponding linear equation
- Inequality symbol or boundary line dashed (not included in solution set)
- Inequality symbol or boundary line solid (included in solution set)
- represents solution set of inequality
- Test point not on boundary line to determine shading
- Test point satisfies inequality shade region containing it
- Test point does not satisfy inequality shade region not containing it
Graphing of linear inequalities
- Convert inequality to : or
- Graph corresponding boundary line
- for strict inequalities ( or )
- for inclusive inequalities ( or )
- Choose test point not on boundary line (0, 0)
- Substitute test point into inequality
- Inequality true shade region containing test point
- Inequality false shade region not containing test point
- Label graph with the inequality
Real-world applications of linear inequalities
- Identify variables and meanings in problem context
- Write inequality representing constraints or conditions in problem
- Pay attention to direction of inequality symbol
- Graph inequality on coordinate plane
- Interpret solution set in problem context
- Company produces two products A and B
- Product A requires 2 hours, product B requires 3 hours
- Maximum 24 hours available for production
- Profit for A is 75
- Write constraint inequality and graph
- Let be number of A and be number of B
- Constraint inequality:
- Graph shows possible combinations of A and B within time constraint
Linear Programming and Feasible Regions
- Linear programming involves optimizing a linear function subject to linear constraints
- Feasible region is the set of all points satisfying all constraints in a linear programming problem
- Graphing multiple linear inequalities on the same coordinate plane creates the feasible region
- The optimal solution is typically found at a vertex of the feasible region
Key Terms to Review (23)
<: The less than symbol, <, is a mathematical operator that indicates a relationship where one value is smaller than another value. It is used in various contexts within algebra to represent inequalities, where the solution set includes all values that satisfy the inequality condition.
>: The greater than symbol (>) is a mathematical operator used to compare two values and indicate that one value is larger than the other. It is a fundamental concept in algebra that is applied in various contexts, including solving linear inequalities, compound inequalities, absolute value inequalities, graphing linear inequalities in two variables, graphing systems of linear inequalities, solving rational inequalities, and solving quadratic inequalities.
≤ (Less Than or Equal To): The symbol '≤' represents the mathematical relationship of 'less than or equal to'. It is used to compare two values and indicate that one value is less than or equal to the other value. This key term is essential in understanding and working with various mathematical concepts, including integers, linear inequalities, compound inequalities, absolute value inequalities, linear inequalities in two variables, systems of linear inequalities, rational inequalities, and quadratic inequalities.
Boundary Line: A boundary line is a conceptual dividing line that separates regions or areas based on certain criteria. In the context of graphing linear inequalities and systems of linear inequalities, the boundary line represents the line that separates the solutions that satisfy the inequality from those that do not.
Coordinate Plane: The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional graphical representation used to locate and visualize points, lines, and other geometric shapes. It consists of a horizontal x-axis and a vertical y-axis that intersect at a point called the origin, forming a grid-like structure that allows for the precise mapping of coordinates.
Dashed Line: A dashed line is a type of line that is composed of a series of short line segments separated by gaps, rather than a continuous unbroken line. This visual representation is used in various mathematical and graphical contexts to convey specific meanings or properties.
Feasible Region: The feasible region is the set of all possible solutions that satisfy a system of linear inequalities in two variables. It represents the area on a coordinate plane where all the constraints or inequalities are met simultaneously.
Greater Than or Equal To (≥): The symbol ≥ is a mathematical operator that represents the relationship where one value is greater than or equal to another value. It is used to compare quantities and express inequalities, indicating that the left-hand side is either greater than or equal to the right-hand side.
Half-plane: A half-plane is a region of the coordinate plane that is divided by a line. It represents all the points on one side of the line, including the line itself. This concept is particularly important in the context of graphing linear inequalities in two variables and systems of linear inequalities.
Linear Inequality: A linear inequality is a mathematical statement that represents an inequality between two linear expressions. It is a type of inequality that involves variables and coefficients in a linear relationship, where the variables are raised to the first power.
Linear Programming: Linear programming is a mathematical optimization technique used to solve problems involving the maximization or minimization of a linear objective function subject to linear constraints. It is a powerful tool for decision-making in various fields, including business, economics, and engineering.
Origin: The origin is a fundamental concept in mathematics, particularly in the context of coordinate systems and graphing. It represents the fixed point of reference from which all other points are measured and located on a graph or coordinate plane.
Quadrant: A quadrant is one of the four equal parts into which a plane or a sphere is divided by two intersecting lines or planes that are perpendicular to each other. It is a fundamental concept in the context of graphing linear inequalities in two variables.
Shaded Region: The shaded region refers to the area on a graph that represents the solution set for a linear inequality or a system of linear inequalities. It is a visual representation of the values that satisfy the given inequality or set of inequalities.
Slope-Intercept Form: The slope-intercept form is a way to represent a linear equation in two variables, $y$ and $x$, in the form $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept, the point where the line crosses the $y$-axis. This form provides a straightforward method for graphing linear equations and understanding their key features.
Solid Line: A solid line is a continuous, unbroken line used in graphing and visual representations to denote specific properties or relationships. In the context of linear inequalities and systems of linear inequalities, the solid line is a key element in accurately depicting the solution set.
Solution Set: The solution set is the set of all values of the variable(s) that satisfy an equation, inequality, or system of equations or inequalities. It represents the collection of all possible solutions to a given mathematical problem.
Standard Form: The standard form of an equation is a specific way of writing the equation that provides a clear and organized structure, making it easier to analyze and work with the equation. This term is particularly relevant in the context of linear equations, quadratic equations, and other polynomial functions.
Test Point Method: The test point method is a graphical technique used to determine the solution set of a linear inequality in two variables. It involves selecting a test point, evaluating the inequality at that point, and using the resulting sign to determine the region that satisfies the inequality.
Two Variables: Two variables are two distinct quantities or characteristics that can change independently within a given context. They are often represented by symbols, such as x and y, and their relationship is the focus of analysis in various mathematical and scientific applications.
X-axis: The x-axis is the horizontal line in a coordinate plane that represents the independent variable or the values along the horizontal dimension. It is the primary reference line for measuring the positions of points along the horizontal direction.
Y-axis: The y-axis is the vertical line on a coordinate plane that represents the vertical or up-and-down dimension. It is used to measure and plot the position of points along the vertical axis of a graph.
Y-intercept: The y-intercept is the point where a line or graph intersects the y-axis, representing the value of the function when the independent variable (x) is equal to zero. It is a crucial concept in understanding the behavior and characteristics of various types of functions and their graphical representations.