Absolute Value Inequalities Practice Problems

Q: How do I rewrite |X| < a as a compound inequality?

For positive a, |X| < a becomes -a < X < a — an 'and' compound giving one interval. ≤ works the same with included endpoints.

Q: How do I rewrite |X| > a?

For positive a, |X| > a becomes X a — an 'or' compound with two separate pieces.

Q: Do I isolate the absolute value first?

Yes. 2|x - 1| - 3 < 7 must become |x - 1| < 5 before translating.

Q: Do I write the answer as an interval or as an inequality?

Either is marked correct. (-∞, -8) ∪ (2, ∞), the ASCII form (-inf,-8) U (2,inf), and the compound inequality x 2 are all accepted for the same problem. Brackets are checked though, so a ≤ in the original has to come back as a square bracket.

Absolute Value Inequalities Practice Problems

The Absolute Value Inequalities lesson explains the “less thAND / greatOR” translation and the special cases; this set is 12 to practice on, each solved in interval notation.

How to use this set

Isolate the absolute value, translate to the matching compound inequality, solve it, and write the answer in interval notation. Type the solution set into the answer box and press Check Answers. Interval notation is accepted in whichever style you write it — (-inf, -8) U (2, inf), (-∞, -8) ∪ (2, ∞), or the inequality form x < -8 or x > 2.

Common mistakes to watch for

Using the wrong compound. Less than → “and” (one interval); greater than → “or” (two pieces).

Writing a “greater than” answer as a single interval.

Not isolating the absolute value first.

Missing the always-true / never-true special cases when the absolute value is compared to a negative.

Where to go next

The compound you produce is solved with the Compound Inequalities Practice Problems. The equation version is the Absolute Value Equations Practice Problems. The answer format is Interval Notation. For the full method, see the Absolute Value Inequalities lesson.

Frequently Asked Questions

How do I rewrite |X| < a as a compound inequality?+

For positive , becomes — an 'and' compound giving one interval. works the same with included endpoints.

How do I rewrite |X| > a?+

For positive , becomes or — an 'or' compound with two separate pieces.

A quick way to remember which is which?+

'Less thAND' — a less-than absolute value gives an AND compound (one interval near zero). 'GreatOR' — a greater-than absolute value gives an OR compound (two pieces far from zero).

What if the absolute value is less than a negative number?+

No solution — an absolute value is never negative. What if it is greater than a negative? All real numbers — an absolute value is always .

Do I isolate the absolute value first?+

Yes. must become before translating.

Do I write the answer as an interval or as an inequality?+

Either is marked correct. (-∞, -8) ∪ (2, ∞), the ASCII form (-inf,-8) U (2,inf), and the compound inequality x < -8 or x > 2 are all accepted for the same problem. Brackets are checked though, so a ≤ in the original has to come back as a square bracket.

📏Absolute Value Inequalities Practice Problems

Absolute Value Inequalities Practice Problems

How to use this set

Common mistakes to watch for

Where to go next

Frequently Asked Questions

More practice problems

Absolute Value Inequalities Practice Problems