Absolute Value Inequalities
Rewrite as a compound inequality.
Key idea
Rewrite as a compound inequality.
In depth
|x| < a means x is within a of zero: −a < x < a. |x| > a means farther than a: x < −a or x > a. Think of it as distance from a centre.
Worked example
|x − 2| ≤ 3 means −3 ≤ x − 2 ≤ 3, so −1 ≤ x ≤ 5.
Formula
Real-world uses
- Setting safe error limits for measuring instruments and rating how close a guess must be
Common mistake
Writing |x| > a as a single interval instead of two pieces.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Absolute Value Inequalities?
Rewrite as a compound inequality. It is a topic at the advanced level in Algebra.
What is the formula for Absolute Value Inequalities?
The key formula is |x| < k ⇒ −k < x < k.
Where is Absolute Value Inequalities used in real life?
Real-world examples: Setting safe error limits for measuring instruments and rating how close a guess must be.
What should I learn before Absolute Value Inequalities?
Make sure you are comfortable with Absolute Value Equations, the previous topic in Algebra.
What is a common mistake with Absolute Value Inequalities?
Writing |x| > a as a single interval instead of two pieces.
How can I practise Absolute Value Inequalities?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Absolute Value Inequalities?
Move on to Coordinate Plane next.
Related advanced topics in Algebra
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.
- Quadratic Formula – Gives roots of any quadratic.