Math July 13, 2026 · 8 Min Read

Absolute Value Inequalities Calculator – Guide & Formulas

Solve absolute value inequalities instantly. Enter |x| < a, |x| > a, |x| <= a, or |x| >= a and get step-by-step interval solutions.

Solve absolute value inequalities with our free calculator. Enter any inequality involving |x| and get interval notation solutions with step-by-step explanations.

Key Takeaway

Use the free Absolute Value Inequalities Calculator to solve absolute value inequalities instantly. enter |x| < a, |x| > a, |x| <= a, or |x| >= a and get step-by-step interval solutions. Get instant results with step-by-step explanations.

How to Use the Absolute Value Inequalities Calculator

  1. Enter the absolute value expression on the left side.
  2. Select the inequality type: <, >, <=, or >=.
  3. Enter the value or expression on the right side.
  4. Review the solution in interval notation and number line representation.

The Formula

|expression| < a means -a < expression < a (AND, intersection). |expression| > a means expression < -a OR expression > a (OR, union). Same logic applies to <= and >=.

Variable Definitions

  • |x|: The absolute value of x
  • a: The boundary value (must be >= 0 for < and <=)
  • <: Less than — solution is inside the interval
  • >: Greater than — solution is outside the interval

Solving |x - 3| < 5

Find all values of x where the absolute value of (x - 3) is less than 5.

  1. Step 1: Convert to compound inequality: -5 < x - 3 < 5.
  2. Step 2: Add 3 to all parts: -5 + 3 < x < 5 + 3.
  3. Step 3: Simplify: -2 < x < 8.
  4. Step 4: In interval notation: (-2, 8).

Frequently Asked Questions

What is the difference between < and <= in absolute value inequalities?

The < symbol means strict inequality (boundary not included, open circle on number line). The <= symbol means inclusive (boundary included, closed circle). The interval notation uses ( ) for strict and [ ] for inclusive.

When do I use AND vs OR for absolute value inequalities?

Use AND (intersection) for "less than" inequalities: |x| < a means -a < x < a. Use OR (union) for "greater than" inequalities: |x| > a means x < -a OR x > a.

What if the right side is negative?

For |x| < negative number, there is no solution (empty set) because absolute value is always >= 0. For |x| > negative number, the solution is all real numbers since absolute value is always >= 0.

How do I write the solution in interval notation?

For |x| < 5: the solution is (-5, 5) meaning all numbers between -5 and 5. For |x| > 5: the solution is (-∞, -5) U (5, ∞) meaning all numbers less than -5 or greater than 5.

Can the absolute value expression be more complex?

Yes. For |2x + 1| <= 7, solve -7 <= 2x + 1 <= 7, which gives -4 <= x <= 3, or in interval notation [-4, 3]. The calculator handles any linear expression inside the absolute value.