Absolute Value Inequalities Calculator
Solve absolute value inequalities step by step
What is an Absolute Value Inequality?
An absolute value inequality is a mathematical statement that involves the absolute value of an expression and an inequality symbol (>), (≥), (<), or (≤). The absolute value of a number represents its distance from zero on a number line, regardless of whether the number is positive or negative.
How to Solve Absolute Value Inequalities
To solve absolute value inequalities, follow these steps:
- Isolate the absolute value expression on one side
- Consider the inequality symbol being used
- For |x| >a: Solve x >a OR x <-a
- For |x| <a: Solve -a <x <a
- Write the solution in interval notation
Types of Absolute Value Inequalities
There are four main types of absolute value inequalities:
- Greater than: |ax + b| >c
- Greater than or equal to: |ax + b| ≥c
- Less than: |ax + b| <c
- Less than or equal to: |ax + b| ≤c
Examples with Solutions
Example 1: |2x + 1| >3
- Solution: x <-2 or x >1
- Interval notation: (-∞, -2) ∪ (1, ∞)
Example 2: |x - 2| <4
- Solution: -2 <x <6
- Interval notation: (-2, 6)
Common Mistakes to Avoid
- Forgetting to consider both positive and negative solutions for greater than inequalities
- Incorrectly applying the inequality symbol when splitting the absolute value
- Not considering the coefficient of x when solving the inequality
- Forgetting that absolute value cannot be negative