How to Graph Inequalities Without Calculator

Introduction

Graphing inequalities is essential for solving real-world problems involving constraints. Unlike equations, inequalities represent ranges of values rather than exact solutions. The process involves:

1. Rewriting the inequality in slope-intercept form
2. Graphing the corresponding line (dashed for strict inequalities)
3. Shading the appropriate region based on the inequality symbol

This method helps visualize all possible solutions to the inequality.

Basic Method for Graphing Inequalities

Step 1: Rewrite in Slope-Intercept Form

Convert the inequality to the form y = mx + b. For example, for 2x + 3y ≤ 6:

Step 2: Graph the Corresponding Line

Plot the y-intercept (0,2) and use the slope (-2/3) to find another point. Use a dashed line for strict inequalities (< or >) and a solid line for non-strict inequalities (≤ or ≥).

Step 3: Shade the Correct Region

Use a test point not on the line to determine which side to shade. For y ≤ (-2/3)x + 2, test (0,0):

Shade below the line.

Worked Example

Graph the inequality x - 2y > 4.

Solution Steps

1. Rewrite in slope-intercept form: y < -0.5x + 2
2. Graph the line y = -0.5x + 2 using a dashed line
3. Test (0,0): 0 < -0.5(0) + 2 → 0 < 2 (True)
4. Shade above the line

The solution is all points above the dashed line.

Common Mistakes to Avoid

• Forgetting to change the inequality symbol when multiplying or dividing by negative numbers
• Using the wrong type of line (solid vs. dashed)
• Shading the incorrect region when testing points
• Not labeling the graph with the inequality