Quick answer

|x| < a means −a < x < a when a > 0. |x| > a means x < −a or x > a when a > 0.

Formula

  • |x| ≤ a → −a ≤ x ≤ a
  • |x| ≥ a → x ≤ −a or x ≥ a

Introduction

Think of |x| < 5 as all numbers within five units of zero, and |x| > 5 as numbers farther than five units away. Distance language keeps AND/OR straight.

Less-than forms produce one bounded interval; greater-than forms split into two rays. The inequality symbol controls which pattern you use.

The Absolute Value Calculator helps test sample values from your interval. For equalities instead of ranges, see absolute value equations. For shading and graphs, read absolute value graph.

Distance language

Less-than inequalities produce a single bounded interval between −a and a when a is positive. Endpoints depend on ≤ or <.

Greater-than inequalities split into two unbounded rays because values far left or far right of zero both have large magnitude.

Shifted forms like |x − 2| < 3 mean x is within three units of 2, not zero. Rewrite as −3 < x − 2 < 3, then solve for x.

Empty solution sets appear when a bound is negative in a less-than problem: no real number can have magnitude less than a negative bound.

Rewrite rules

  • |x| &lt; a → −a &lt; x &lt; a (a &gt; 0)
  • |x| ≤ a → −a ≤ x ≤ a (a &gt; 0)
  • |x| &gt; a → x &lt; −a or x &gt; a (a &gt; 0)
  • |x| ≥ a → x ≤ −a or x ≥ a (a &gt; 0)

Compound inequalities connect directly to graphing: strict less-than corresponds to the interior of the V below a horizontal line.

Use AND between bounds for less-than type. Use OR between outside regions for greater-than type. Mixing them is the most common mistake.

Algebra courses combine these rules with case splits, graphs, and equation solving in the same chapter.

Inequality method

  1. Identify the form. Decide whether the inequality is less-than type or greater-than type before you rewrite.
  2. Remove the bars. Write the equivalent compound inequality or two separate inequalities.
  3. Solve for x. Isolate x with ordinary algebra rules. Add or subtract the same amount on all three parts when needed.
  4. Graph or test. Shade the interval and pick test points, including endpoints when ≤ or ≥ appears.

Sample inequalities

Solve |x − 2| ≤ 3. This means −3 ≤ x − 2 ≤ 3, so −1 ≤ x ≤ 5.

Solve |x| > 4. This means x < −4 or x > 4.

Solve |2x + 1| < −2. No solution because |2x + 1| ≥ 0 cannot be less than a negative number.