Graphing Inequalities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Graphing Inequalities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Graphing inequalities represents all solution values on a number line or coordinate plane.

Use boundary lines and shading to show where conditions are true.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A graph of an inequality shows the entire solution region — a shaded half-plane or interval, not just one point.

Common stuck point: Students shade the wrong side of the boundary — always test a point to determine which side satisfies the inequality.

Sense of Study hint: Test one point not on the boundary (often the origin) to choose the correct region.

Worked Examples

Example 1

easy
Graph x > 2 on a number line.

Solution

  1. 1
    Place an open circle at x = 2 (not included, strict inequality).
  2. 2
    Shade everything to the right of 2.
  3. 3
    The arrow extends to +\infty.

Answer

Open circle at 2, shaded right → (2, \infty).
On a number line, open circles represent strict inequalities (<, >) and closed circles represent inclusive inequalities (\leq, \geq). Shade in the direction of the solution.

Example 2

medium
Graph the solution of y \leq 2x + 1 on the coordinate plane.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Graph x \leq -1 on a number line.

Example 2

medium
Is the point (1, 3) in the solution region of y < x + 4?
← Back to Graphing Inequalities Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

inequalitiescoordinate planenumber line