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Math Practice

Absolute Value Equations Practice Problems

Practice solving absolute value equations: isolate the absolute value, then split |X| = a into X = a and X = −a. Every equation has two integer solutions and asks for the larger one.

  1. 1.

    Solve \(2\left| x + 8 \right| = 20\). Enter the larger solution.

  2. 2.

    Solve \(2\left| x - 4 \right| = 10\). Enter the larger solution.

  3. 3.

    Solve \(2\left| x + 6 \right| = 8\). Enter the larger solution.

  4. 4.

    Solve \(4\left| x - 1 \right| = 40\). Enter the larger solution.

  5. 5.

    Solve \(3\left| x - 8 \right| = 15\). Enter the larger solution.

  6. 6.

    Solve \(3\left| x + 1 \right| = 33\). Enter the larger solution.

  7. 7.

    Solve \(4\left| x - 4 \right| = 16\). Enter the larger solution.

  8. 8.

    Solve \(4\left| x - 4 \right| = 8\). Enter the larger solution.

  9. 9.

    Solve \(2\left| x - 6 \right| = 14\). Enter the larger solution.

  10. 10.

    Solve \(4\left| x - 3 \right| = 20\). Enter the larger solution.
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Mathovia Team
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Absolute Value Equations Practice Problems

The Absolute Value Equations lesson covers the split into two cases and every special case; this practice set drills the standard two-solution problem with instant checking. Each asks for the larger solution.

How the difficulty levels work

DifficultyWhat it drillsExample
Easy, already isolated
Medium — divide first
Hard — subtract, then divide

Easy starts with the absolute value already alone: split, solve both linear equations, take the larger root.

Medium puts a coefficient in front. Divide both sides by it to isolate the absolute value, then split.

Hard adds a constant outside the bars. Subtract it, then divide, then split.

Using the practice problems

Pick a difficulty and a count. For each equation, isolate the absolute value, split into two cases, solve both, and enter the larger solution. Then:

  • Check Answers grades the set.
  • Show Answers reveals every pair of solutions.
  • Print Worksheet builds a printable page.

Write both solutions on paper even though you enter one — problems where both matter come next, in the inequalities.

The rules these practice problems drill

RuleStatement
Isolate firstget alone before splitting
The split () or
Each caseis a linear equation, solved normally
Two solutionsreport the larger here

See the Absolute Value Equations lesson for the zero and negative cases and for .

Common mistakes to watch for

Splitting before isolating. is not ; get first.

Writing only one case. A positive right side always gives two equations.

Sign slip in case two. is , not .

Mixing up which root is larger when both are negative — is larger than .

Where to go next

The inequality version — where both solutions become boundaries of an interval — is the Absolute Value Inequalities Practice Problems. Each split case is an ordinary linear equation; drill those with the Algebra Equation Practice Problems. For every special case, see the Absolute Value Equations lesson.

Frequently Asked Questions

Why does the problem ask for the larger solution?+

An absolute value equation with a positive right side has two solutions, and the answer box checks one number. Asking for the larger root confirms you split into both cases and solved each correctly.

How do I enter my answer?+

Type the larger of the two solutions as a plain integer — for example 11, or -3. You can also write x = 11.

How do I solve |X| = a?+

For a positive , split into two equations: and . Solve both. Here is the expression inside the bars, so gives or , i.e. or .

What changes at the Medium and Hard levels?+

Medium multiplies the absolute value by a constant, so you divide first: becomes . Hard also adds a constant outside, so you subtract before dividing: .

Do I always isolate the absolute value before splitting?+

Yes. Splitting directly gives wrong equations. Get first, then split.

Are these ever no-solution problems?+

Not in this set — every generated equation isolates to a positive number, so there are always two solutions. The no-solution case (absolute value equal to a negative) and the one-solution case (equal to zero) are covered in the lesson.

More practice problems

Absolute Value Inequalities Practice Problems Algebra Equation Practice Problems Linear Inequalities Practice Problems