Systems of Linear Inequalities

Definitions and Learning Outcomes

  • Solutions to Systems of Linear Inequalities: Identify regions in the coordinate plane that satisfy multiple inequalities.

  • Graphing a System of Two Inequalities: Learn to plot multiple inequalities on the same graph and identify their intersections.

Graphing a Linear Inequality

  • A single linear inequality divides the coordinate plane into two regions:

    • Solution Region: Points that satisfy the inequality.

    • Non-Solution Region: Points that do not satisfy the inequality.

  • Example: For the inequality y < 2x + 5, the shaded region below the dashed line represents all solutions.

    • The line is dashed since points on the line do not satisfy this strict inequality.

Graphing Multiple Inequalities

  • When graphing systems of inequalities, e.g., y < 2x + 5 and y > -x:

    • Shaded Regions: Shade below the dashed line for y < 2x + 5 and above the dashed line for y > -x.

    • Solution to System: The purple region, where both shaded areas overlap, contains all solutions to the system of inequalities.

Testing for Solutions in a System

  • To check if a point (e.g., (2, 1)) is a solution for a system:

    1. Substitute the x and y coordinates into each inequality.

    2. Verify if the inequalities hold true:

    • For x + y > 1:
      2 + 1 > 1
      TRUE

    • For 2x + y < 8:

    • 2(2) + 1 < 8 \rightarrow 4 + 1 < 8 \rightarrow \text{TRUE} once both inequalities are true, (2, 1) is a solution for the system.

Example of a Non-Solution

  • If point (2, 1) fails any inequality in the system, it is not a solution. For example:

    • For 3x + y < 4:
      3(2) + 1 < 4
      6 + 1 < 4
      —> FALSE

    • Thus, (2, 1) is not a solution for this system despite being a solution for one inequality.

Systems with No Solutions

  • If the boundaries of two inequalities are parallel (same slope), then:

    • There is no intersection; hence no solutions exist.

    • Example: Graphing y < 2x - 3 and y \text{ (solid line)} \text{ vs. } y < 2x + 1: both inequalities are parallel, and the shaded regions do not overlap.

Summary

  • Solutions of a System: Regions of points satisfying all inequalities in the system.

  • Verification Method: The same method applies for both systems of linear equations and inequalities.

  • No Solutions: Occurs when lines are parallel and do not intersect, resulting in no shared shaded region for the inequalities.