Solving Absolute Value Inequalities in Interval Notation Calculator

Absolute value inequalities are fundamental in algebra and calculus. This guide explains how to solve them in interval notation, with a calculator to simplify the process.

Introduction

Absolute value inequalities compare the absolute value of an expression to a number. They're written in the form |x| < a, |x| > a, etc. Solving them requires understanding the definition of absolute value and applying it to inequalities.

Interval notation provides a concise way to represent solutions. For example, x > 3 can be written as (3, ∞).

Basic Formulas

The absolute value of a number is its distance from zero on the number line, regardless of direction. Mathematically:

|x| = x if x ≥ 0
-x if x < 0

When solving inequalities, we consider two cases based on the definition above.

Solving Method

To solve |x| < a:

Rewrite as -a < x < a
Express in interval notation: (-a, a)

For |x| > a:

Rewrite as x < -a or x > a
Express in interval notation: (-∞, -a) ∪ (a, ∞)

Remember to reverse the inequality sign when multiplying or dividing by negative numbers.

Examples

Example 1: |x| < 5

Solution: -5 < x < 5 → (-5, 5)

Example 2: |3x + 2| > 4

Solution: 3x + 2 < -4 or 3x + 2 > 4 → x < -2 or x > -2/3 → (-∞, -2) ∪ (-2/3, ∞)

Common Mistakes

Forgetting to consider both cases when solving absolute value inequalities
Incorrectly reversing inequality signs when multiplying/dividing by negatives
Misinterpreting interval notation (e.g., [a, b] vs. (a, b))

FAQ

What is interval notation?: Interval notation is a way to represent sets of real numbers using parentheses and brackets. Parentheses indicate endpoints are not included, while brackets indicate they are.
How do I solve |x - a| < b?: Rewrite as -b < x - a < b, then add a to all parts: a - b < x < a + b → (a - b, a + b)
What does the solution to |x| > 0 mean?: It means all real numbers except zero: (-∞, 0) ∪ (0, ∞)