Graph Absolute-Value Inequalities: Boundary, Test Points, and Shading

Algebra and Function Practice

Absolute-value inequalities describe distances from a center. Graphing them means finding the boundary lines and deciding which side to shade. This lesson teaches a reliable three-step method: rewrite, graph the boundary, then test a point. You will work with inequalities such as |x| < 3, |y| ≥ 2, and |x - 1| ≤ 4. The boundary is always a pair of parallel lines (vertical or horizontal) or a V-shape when both variables appear inside the absolute value. For this lesson, we focus on inequalities with one variable inside the absolute value, which produce vertical or horizontal boundary lines. The shading convention is explicit: shade the region where the inequality is true, and use a dashed boundary for strict inequalities (<, >) and a solid boundary for inclusive inequalities (≤, ≥). A test point confirms the correct side. All numerical examples are hypothetical and chosen for easy arithmetic.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Rewrite the absolute-value inequality

An absolute-value inequality like |x| < 3 means the distance from x to 0 is less than 3. This rewrites as -3 < x < 3, a compound inequality. Similarly, |x| ≥ 2 rewrites as x ≤ -2 or x ≥ 2. The key is to split into two cases: one with the expression as written, and one with the expression negated and the inequality direction flipped for the 'or' case.

For |x - h| < r, rewrite as h - r < x < h + r. For |x - h| > r, rewrite as x < h - r or x > h + r. The same logic applies to y. Always check whether the inequality is strict (<, >) or inclusive (≤, ≥), because that determines whether the boundary lines are dashed or solid.

1. Identify the center h and the radius r from the absolute-value expression.
2. For < or ≤, write a single compound inequality: h - r < x < h + r (or ≤).
3. For > or ≥, write two separate inequalities joined by 'or': x < h - r or x > h + r (or ≤/≥).
4. Label the boundary values on the number line or coordinate grid.

Graph the boundary lines

For a one-variable absolute-value inequality, the boundary consists of two vertical lines (if the variable is x) or two horizontal lines (if the variable is y). For example, |x| < 3 has boundary lines x = -3 and x = 3. Because the inequality is strict (<), both lines are dashed. For |y| ≥ 2, the boundary lines are y = -2 and y = 2, and both are solid because the inequality is inclusive (≥).

Plot the boundary lines accurately using the grid. Label each line with its equation. The region between or outside these lines will be shaded after testing.

1. Draw vertical lines for x-boundaries or horizontal lines for y-boundaries.
2. Use dashed lines for < and >; use solid lines for ≤ and ≥.
3. Label each boundary line with its equation, such as x = -3 or y = 2.
4. Double-check that the boundary values match your rewritten compound inequality.

Choose and apply a test point

Pick a test point that is not on any boundary line. Substitute its coordinates into the original absolute-value inequality. If the result is true, shade the region containing that test point. If false, shade the opposite region. For inequalities with 'or', the true region may be two separate areas; shade both.

A convenient test point is often (0,0) if it is not on a boundary. For |x| < 3, test (0,0): |0| < 3 is true, so shade the strip between x = -3 and x = 3. For |y| ≥ 2, test (0,0): |0| ≥ 2 is false, so shade above y = 2 and below y = -2, not the middle strip.

1. Select a test point not on any boundary line, such as (0,0) when possible.
2. Substitute the point into the original inequality and evaluate.
3. If true, shade the region containing the test point; if false, shade the other region(s).
4. For 'or' inequalities, check both outside regions and shade both if true.

Check your graph

Verify your shading by testing a second point in the shaded region and a point in the unshaded region. The shaded point should make the inequality true; the unshaded point should make it false. Also confirm that boundary lines are dashed or solid correctly. A common error is shading the wrong side after a false test, or forgetting to shade both outside regions for 'or' inequalities.

Keep units consistent: if the grid represents units, state that each square is one unit. All numerical values in this lesson are hypothetical examples for practice.

1. Test one point inside your shaded region; it must satisfy the inequality.
2. Test one point outside your shaded region; it must not satisfy the inequality.
3. Confirm dashed versus solid boundary lines match strict versus inclusive inequalities.
4. For 'or' inequalities, confirm both separate regions are shaded.

Worked example

Graph the inequality |x - 1| ≤ 4. Rewrite as a compound inequality, graph the boundary, and shade using a test point.

Four practice exercises

Common mistakes

• Shading between the boundary lines for a 'greater than' absolute-value inequality. For |x| > r, the true region is outside the interval. Test (0,0): if false, shade the two outer regions, not the middle.
• Using solid boundary lines for strict inequalities (< or >). Strict inequalities exclude the boundary, so use dashed lines. Inclusive inequalities (≤ or ≥) use solid lines.
• Forgetting to flip the inequality direction when rewriting the negative case. For |x| < 3, the compound is -3 < x < 3. For |x| > 3, it is x < -3 or x > 3. Write both cases explicitly before graphing.
• Testing a point that lies on a boundary line. Choose a test point clearly inside a region, such as (0,0) when it is not on a boundary. A boundary point makes the inequality equal, which does not decide shading.

Self-check rubric

• Rewrites the absolute-value inequality correctly as a compound inequality or two separate inequalities.
• Identifies both boundary values and draws them as vertical or horizontal lines.
• Uses dashed lines for strict inequalities and solid lines for inclusive inequalities.
• Selects a test point not on a boundary and substitutes correctly.
• Shades the region that makes the inequality true, including both outer regions for 'or' inequalities.
• Labels boundary lines and indicates the shaded region clearly.
• Accepts valid alternative test points and equivalent shading descriptions, provided the region is correct.

Questions and answers