How to Graph Absolute Value Inequalities?

Graphing absolute value inequalities on the coordinate plane combines two skills: graphing the related absolute value equation (the boundary V) and determining which region to shade. The result is a shaded half-plane on one or both sides of the V-shaped boundary. This guide walks through the process step by step, with worked examples, two video lessons, and practice problems.

What Is an Absolute Value Inequality on a Graph?

An absolute value inequality in two variables, such as \(\color{blue}{y < |x|}\) or \(\color{blue}{y \ge |x - 2| + 1}\), defines a region of the coordinate plane. The boundary is the V-shaped graph of the related equation. Whether that boundary is solid or dashed depends on whether the inequality includes “equal to” (≤ or ≥) or not (< or >).

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How to Graph an Absolute Value Inequality

Step 1: Graph the Boundary

Replace the inequality symbol with an equals sign and graph the absolute value equation. This gives you the V-shaped boundary.

• Use a solid line (or solid V) for ≤ or ≥.
• Use a dashed line (or dashed V) for < or >.

Step 2: Identify the Vertex and Arms

Write the equation in the form \(\color{blue}{y = a}\)|\(\color{blue}{x – h}\)| + k. The vertex is (h, k). Determine whether the V opens up (a > 0) or down (a < 0).

Step 3: Test a Point

Choose a test point NOT on the boundary (the origin (0, 0) works unless it is on the boundary). Substitute it into the inequality:

• If the inequality is true, shade the region containing the test point.
• If the inequality is false, shade the opposite region.

Shading Shortcut

• y < |expression| or y ≤ |expression|: shade below (and outside) the V.
• y > |expression| or y ≥ |expression|: shade above (inside) the V.

Step-by-Step Summary

1. Write the related equation by replacing the inequality symbol \(\color{blue}{\text{ with } =}\).
2. Find the vertex (h, k) and the direction the V opens.
3. Draw the boundary: solid for ≤ or ≥; dashed for < or >.
4. Test a point not on the boundary in the original inequality.
5. Shade the region where the inequality is true.

Watch: Graphing Absolute Value Inequalities (Video Lesson)

Khan Academy demonstrates how to graph absolute value inequalities on the coordinate plane:

Graphing Absolute Value Inequalities – Worked Examples

Example 1: Graph \(\color{blue}{y < |x|}\)

Boundary: y = |x|; vertex (0, 0); dashed V (strict <).
Test (0, −1): −1 < |0| = 0 → −1 < 0 ✓ (true)
Shade below the dashed V.

Example 2: Graph \(\color{blue}{y \ge |x – 2| + 1}\)

Boundary: y = |\(\color{blue}{x – 2}\)| + 1; vertex (2, 1); solid V (includes equal to).
Test (2, 5): 5 ≥ |\(\color{blue}{2-2}\)|+\(\color{blue}{1 = 1}\) ✓ (true)
Shade above and on the solid V (inside the V arms).

Example 3: Graph \(\color{blue}{y \le -|x + 1| + 4}\)

Boundary: y = −|\(\color{blue}{x + 1}\)| + 4; vertex (−1, 4); opens down; solid V.
Test (0, 0): 0 ≤ −|\(\color{blue}{0+1}\)|+4 = −\(\color{blue}{1+4 = 3}\) ✓ (true)
Shade below and on the solid downward V.

Example 4: Graph \(\color{blue}{y > 2|x – 3| – 2}\)

Boundary: \(\color{blue}{y = 2}\)|\(\color{blue}{x-3}\)|−2; vertex (3, −2); dashed V, steeper arms.
Test (3, 0): 0 > 2|\(\color{blue}{3-3}\)|−2 = −2 ✓ (true)
Shade above the dashed V (the interior region).

More Practice: Solving and Graphing Absolute Value Inequalities (Video Lesson)

Math and Science provides an in-depth walkthrough of both solving and graphing absolute value inequalities:

Exercises: Graphing Absolute Value Inequalities

1. Graph \(\color{blue}{y > |x + 3|}\). State the vertex and the shaded region.
2. Graph \(\color{blue}{y \le |x| – 2}\). Is the boundary solid or dashed?
3. What is the vertex of the boundary for \(\color{blue}{y < -|x - 1| + 5}\)?
4. Describe the shaded region for \(\color{blue}{y \ge 3|x|}\).
5. For \(\color{blue}{y < |2x - 4| + 1}\), test the point (2, 0). Is it in the solution region?
6. Write an absolute value inequality whose graph has vertex (0, 3), opens up, with shading above the boundary.

Answers

1. Vertex (−3, 0); dashed V; shade above (inside) the V.
2. Solid boundary; shade below (outside) the V; vertex (0, −2).
3. Vertex (1, 5).
4. Above and on the V; vertex (0, 0); solid boundary (includes =).
5. y = |\(\color{blue}{2(2)-4}\)|+1 = |0|+\(\color{blue}{1 = 1}\); test: 0 < 1 → true, so (2, 0) is in the solution region.
6. y ≥ |x| + 3 (or any y ≥ a|\(\color{blue}{x-h}\)|+3 with a > 0 and vertex at (0,3)).

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Free Graphing Absolute Value Inequalities Worksheet

Ready to practice on your own? Download our free Graphing Absolute Value Inequalities worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Graphing Absolute Value Inequalities before a quiz or test.

Download Absolute Value Inequalities Worksheet

Frequently Asked Questions

When is the boundary of an absolute value inequality drawn as a dashed line?

Use a dashed boundary when the inequality is strict (< or >). A dashed line means points on the boundary are not included in the solution. Use a solid boundary for ≤ or ≥, which means the boundary points are included.

How do I decide which side of the V to shade?

Always use a test point that is not on the boundary. Substitute its coordinates into the original inequality. If the result is true, shade the side containing the test point; if false, shade the other side.

What is the difference between graphing a one-variable and a two-variable absolute value inequality?

A one-variable absolute value inequality (e.g., |x| < 3) is graphed on a number line and produces a segment or two rays. A two-variable absolute value inequality (e.g., y < |x|) is graphed on the coordinate plane and produces a shaded region bounded by a V-shaped curve.

Related Topics

• Solving Absolute Value Inequalities
• Graphing Absolute Value Equations
• Solving Absolute Value Equations
• Graphing Linear Inequalities
• Piecewise Functions