Solving Linear Inequalities Including Graphing Techniques

Unit: Linear inequalities in one or two variables

Solving Linear Inequalities Including Graphing Techniques
Overview

Linear inequalities are mathematical statements that compare two expressions using inequality symbols such as <, >, ≤, or ≥. In one variable, linear inequalities produce a solution set representing a range of values that satisfy the inequality. In two variables, they define regions of the coordinate plane.

Solving Linear Inequalities in One Variable

1. Solving and Graphing:
   • Treat the inequality like an equation when solving.
   • Graph the solution set on a number line.
   • Use an open circle for < and > and a closed circle for ≤ and ≥.
   • Draw an arrow indicating the interval where the inequality holds true.
2. Examples:
   • 2x−3<5
     • Solve: 2𝑥<5+3, x<4
     • Graph: Open circle at 4, arrow pointing left.
   • 3−x≥7
     • Solve: 3−𝑥≥7, −𝑥≥4, 𝑥≤−4
     • Graph: Closed circle at -4, arrow pointing right.

Solving Linear Inequalities in Two Variables

1. Graphing Technique:
   • Treat the inequality as an equation and graph the corresponding line.
   • Determine if the region above or below the line (or to the left or right) satisfies the inequality.
   • Use a dashed line for < or > and a solid line for ≤ or ≥.
   • Test a point in the region to determine shading (e.g., if (0,0) satisfies the inequality, shade that side of the line).
2. Examples:
   • 2𝑥+3𝑦<6
     • Graph: Plot 2𝑥+3𝑦=6 (dashed line), test a point (e.g., (0,0)), and shade below the line.
   • 3𝑥−2𝑦≥4
     • Graph: Plot 3x−2y=4 (solid line), test a point (e.g., (0,0)), and shade above the line.

Systems of Linear Inequalities

1. Graphical Technique:
   • Graph each inequality separately.
   • The solution is the overlapping region of all shaded areas.
2. Examples:

• Graph: Plot x + y=4 (shaded below), 2x−y=1 (shaded above), overlapping shaded area is the solution.

Summary

• Linear inequalities in one variable produce solution sets on number lines, while in two variables, they define shaded regions in the coordinate plane.
• Solving involves treating the inequality as an equation and graphing the solution set or shaded region.
• Systems of linear inequalities can be solved graphically by finding the overlapping shaded regions.