How to Graph an Inequality: A Step‑by‑Step Guide
2025-10-24 - 17:01
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Understanding the Basics
Before you pick up a pencil, it’s important to know what an inequality represents. Unlike an equation, which shows a single line of points that satisfy the statement, an inequality (such as y ≤ 2x + 3) defines a region on the coordinate plane. This region includes all points that make the statement true.
Step 1: Convert the Inequality to Slope‑Intercept Form
Most linear inequalities can be rewritten as y = mx + b. Doing this helps you identify the slope (m) and the y‑intercept (b) quickly. For example, 3x – 2y > 6 becomes y < (3/2)x – 3 after isolating y.
Step 2: Draw the Boundary Line
Plot the line that corresponds to the equality part of the inequality (y = mx + b). Use a:
- Solid line if the inequality includes ≤ or ≥ (the boundary is part of the solution).
- Dashed line if the inequality uses < or > (the boundary is not included).
Mark the intercepts first, then use the slope to locate a second point, and draw the line through them.
Step 3: Choose a Test Point
Select any point not on the boundary—commonly the origin (0,0) works unless the line passes through it. Substitute the test point into the original inequality:
- If the statement is true, shade the region that contains the test point.
- If false, shade the opposite side of the line.
For y ≤ 2x + 3, plugging (0,0) gives 0 ≤ 3, which is true, so shade the area below the line.
Step 4: Verify and Label
Double‑check a few additional points to ensure the correct side is shaded. Adding a label such as “y ≤ 2x + 3” near the shaded region helps readers understand the solution set at a glance.
Tips for Graphing Non‑Linear Inequalities
When dealing with quadratic or absolute‑value inequalities, plot the corresponding curve first, then test points to decide which region to shade. Remember to use a solid curve for ≤ or ≥ and a dashed curve for < or >.
Conclusion
Graphing an inequality is a straightforward process once you identify the boundary, choose the correct line style, and shade the appropriate region. By following these steps, you’ll be able to visualize solution sets quickly and accurately—an essential skill for algebra, calculus, and beyond.
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