Graphing Linear Inequalities in Two Variables 2.4

Graphing Linear Inequalities in Two Variables

Introduction

Linear inequalities in two variables are mathematical statements that compare linear expressions using inequality symbols. The solution to such an inequality is a region of the coordinate plane, often represented visually by shading. Understanding how to graph these inequalities is essential for solving systems and interpreting solution sets in College Algebra.

Steps for Graphing a Linear Inequality

1. Solve the inequality for y

   • Rewrite the inequality so that y is isolated on one side. This makes it easier to graph and interpret.

2. Graph the related equation (the boundary line)

   • Replace the inequality symbol with an equals sign to get the boundary line equation.

   • Solid line: Use a solid line if the inequality is ≤ or ≥. Points on the line are included in the solution set.

   • Dashed line: Use a dashed line if the inequality is < or >. Points on the line are not included in the solution set.

3. Shade the appropriate region

   • Shade above the line if the inequality is > or ≥.

   • Shade below the line if the inequality is < or ≤.

Key Terms and Concepts

• Linear Inequality: An inequality that can be written in the form Ax + By < C, Ax + By ≤ C, Ax + By > C, or Ax + By ≥ C.

• Boundary Line: The line corresponding to the related equation (with = instead of the inequality symbol). It divides the plane into two regions.

• Solution Region: The set of all points (x, y) that satisfy the inequality, represented by the shaded area.

Example 1: Graphing 3x + 2y ≥ 6

• Step 1: Solve for y:

  • Subtract 3x from both sides: $2y \geq -3x + 6$

  • Divide both sides by 2: $y \geq -\frac{3}{2}x + 3$

• Step 2: Graph the boundary line $y = -\frac{3}{2}x + 3$

  • Since the inequality is ≥, use a solid line.

  • Slope: $-\frac{3}{2}$

  • Y-intercept: (0, 3)

• Step 3: Shade the region above the line (since the inequality is ≥).

• Check: Test the point (2, 0):

  • $3(2) + 2(0) \geq 6$

  • $6 + 0 \geq 6$

  • $6 \geq 6$ (True)

Example 2: Graphing x - 3 < 1

• Step 1: Solve for x:

  • Add 3 to both sides: $x < 4$

• Step 2: Graph the boundary line $x = 4$

  • Since the inequality is <, use a dashed line.

• Step 3: Shade the region to the left of the line (since the inequality is <).

Summary Table: Boundary Line and Shading Rules

Inequality Symbol	Boundary Line	Shading
< or >	Dashed	Below (for <), Above (for >)
≤ or ≥	Solid	Below (for ≤), Above (for ≥)

Additional info:

• When checking a solution, substitute a test point (often (0,0) if not on the boundary) into the original inequality to determine which region to shade.

• These techniques are foundational for solving systems of inequalities and for applications in optimization problems.