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Math Practice

Absolute Value Inequalities Practice Problems

Twelve hand-built absolute value inequalities with full worked solutions. Each isolates the absolute value, translates to the right compound inequality ('and' for less than, 'or' for greater than), and reports the answer in interval notation.

  1. 1.

    Solve \(|x| < 7\).

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    \(-7 < x < 7\). Answer: \((-7, 7)\).

  2. 2.

    Solve \(|x| \ge 3\).

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    \(x \le -3\) or \(x \ge 3\). Answer: \((-\infty, -3] \cup [3, \infty)\).

  3. 3.

    Solve \(|x - 2| \le 6\).

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    \(-6 \le x - 2 \le 6 \Rightarrow -4 \le x \le 8\). Answer: \([-4, 8]\).

  4. 4.

    Solve \(|2x - 1| < 9\).

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    \(-9 < 2x - 1 < 9 \Rightarrow -8 < 2x < 10 \Rightarrow -4 < x < 5\). Answer: \((-4, 5)\).

  5. 5.

    Solve \(|x + 3| > 5\).

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    \(x + 3 < -5\) or \(x + 3 > 5 \Rightarrow x < -8\) or \(x > 2\). Answer: \((-\infty, -8) \cup (2, \infty)\).

  6. 6.

    Solve \(|3x + 2| \ge 8\).

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    \(3x + 2 \le -8\) or \(3x + 2 \ge 8 \Rightarrow x \le -\tfrac{10}{3}\) or \(x \ge 2\).

    Answer: \(\left(-\infty, -\dfrac{10}{3}\right] \cup [2, \infty)\).

  7. 7.

    Solve \(|5 - x| \ge 7\).

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    \(5 - x \le -7\) or \(5 - x \ge 7 \Rightarrow x \ge 12\) or \(x \le -2\).

    Answer: \((-\infty, -2] \cup [12, \infty)\).

  8. 8.

    Solve \(|4x - 3| > 5\).

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    \(4x - 3 < -5\) or \(4x - 3 > 5 \Rightarrow x < -\tfrac{1}{2}\) or \(x > 2\).

    Answer: \(\left(-\infty, -\dfrac{1}{2}\right) \cup (2, \infty)\).

  9. 9.

    Solve \(|x - 4| < -2\).

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    An absolute value can't be less than a negative. Answer: \(\varnothing\).

  10. 10.

    Solve \(|x + 1| > -4\).

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    An absolute value is always \(\ge 0 > -4\). Answer: \((-\infty, \infty)\).

  11. 11.

    Solve \(2|x - 1| + 3 \le 11\).

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    \(2|x - 1| \le 8 \Rightarrow |x - 1| \le 4 \Rightarrow -4 \le x - 1 \le 4 \Rightarrow -3 \le x \le 5\). Answer: \([-3, 5]\).

  12. 12.

    Solve \(3|x + 2| - 1 < 8\).

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    \(3|x + 2| < 9 \Rightarrow |x + 2| < 3 \Rightarrow -3 < x + 2 < 3 \Rightarrow -5 < x < 1\). Answer: \((-5, 1)\).
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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

These problems are hand-written, not auto-generated. Isolate the absolute value, translate to the matching compound inequality, solve it, and write the answer in interval notation. Then compare with the solution.

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.

Why is this a curated set?+

The answers are intervals and unions, not single numbers. Each problem has a full worked solution.

More practice problems

Absolute Value Equations Practice Problems Compound Inequalities Practice Problems Interval Notation Practice Problems