5.10 Systems of Linear Inequalities in Two Variables
2 min read•Last Updated on June 18, 2024
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.
Understanding 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.
Systems of Linear Inequalities in Two Variables
Graphing linear inequality systems
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Linear inequality in two variables takes the form ax+by<c, ax+by≤c, ax+by>c, or ax+by≥c, where a, b, and c are real numbers and a and b are not both zero
Inequality sign determines which half-plane is shaded
Dashed boundary line used for strict inequalities < or > (y<2x+1)
Solid boundary line used for inclusive inequalities ≤ or ≥ (y≤−3x+4)
Graphing a system of linear inequalities involves:
Graph each inequality separately on the same coordinate plane
Solution set is the intersection of all shaded regions satisfying the inequalities
Solution set of a system of linear inequalities represents all ordered pairs (x,y) that simultaneously satisfy every inequality in the system
Ordered pairs in inequality systems
Checking if an ordered pair (x,y) satisfies a system of inequalities:
Substitute x and y values into each inequality in the system
If substitution results in true statements for all inequalities, the ordered pair is a solution
Ordered pair satisfying all inequalities in the system is a solution point contained within the solution set (graphically represented by the intersecting shaded regions)
Real-world applications of inequalities
Systems of linear inequalities model real-world situations involving constraints or limitations
Resource allocation problems (budget constraints)
Production limitations (manufacturing capacity)
Feasible regions in optimization problems (maximizing profit)
Modeling a real-world scenario with inequalities:
Identify variables and define them in the context of the problem
Formulate inequalities representing constraints or limitations on the variables
Graph the system of inequalities to visualize the feasible solution region
Interpret the solution set in terms of the original real-world problem
Linear programming problems involve optimizing an objective function subject to constraints modeled by a system of linear inequalities (maximizing profit while satisfying production constraints)
Key components of linear inequalities
Coordinate plane: The 2D plane where linear inequalities are graphed, consisting of x and y axes
Quadrants: The four regions of the coordinate plane, divided by the x and y axes
Slope: The rate of change in a linear equation, representing the steepness of the line
Y-intercept: The point where a line crosses the y-axis, often used as a starting point when graphing
Feasibility: The property of a solution satisfying all constraints in a system of linear inequalities