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Math Practice

Compound Inequalities Practice Problems

Twelve hand-built compound inequalities with full worked solutions. Each shows whether the result is an intersection ('and', one interval) or a union ('or', two pieces), and the final answer in interval notation.

  1. 1.

    Solve \(-4 < x + 2 < 5\).

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    Subtract 2 from all parts: \(-6 < x < 3\). Answer: \((-6, 3)\).

  2. 2.

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

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    Add 1: \(4 \le 2x \le 12\). Divide by 2: \(2 \le x \le 6\). Answer: \([2, 6]\).

  3. 3.

    Solve \(x < 6\) or \(x > 1\).

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    The two pieces cover the whole line. Answer: \((-\infty, \infty)\).

  4. 4.

    Solve \(x - 3 > 2\) and \(x + 1 < 10\).

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    \(x > 5\) and \(x < 9\). Overlap: \((5, 9)\).

  5. 5.

    Solve \(x + 2 \le -3\) or \(x - 4 \ge 1\).

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    \(x \le -5\) or \(x \ge 5\). Answer: \((-\infty, -5] \cup [5, \infty)\).

  6. 6.

    Solve \(-6 \le \dfrac{x}{2} < 4\).

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    Multiply all parts by 2: \(-12 \le x < 8\). Answer: \([-12, 8)\).

  7. 7.

    Solve \(-9 \le 5x + 1 < 16\).

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    Subtract 1: \(-10 \le 5x < 15\). Divide by 5: \(-2 \le x < 3\). Answer: \([-2, 3)\).

  8. 8.

    Solve \(x + 5 \le 2\) and \(3x \ge -12\).

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    \(x \le -3\) and \(x \ge -4\). Overlap: \([-4, -3]\).

  9. 9.

    Solve \(4 < 7 - x \le 10\).

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    Subtract 7: \(-3 < -x \le 3\). Divide by \(-1\) (flip): \(3 > x \ge -3\), i.e. \(-3 \le x < 3\). Answer: \([-3, 3)\).

  10. 10.

    Solve \(2x \ge 8\) and \(x < 3\).

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    \(x \ge 4\) and \(x < 3\): no overlap. Answer: \(\varnothing\).

  11. 11.

    Solve \(3x - 2 < -11\) or \(2x + 1 > 9\).

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    \(3x < -9 \Rightarrow x < -3\), or \(2x > 8 \Rightarrow x > 4\). Answer: \((-\infty, -3) \cup (4, \infty)\).

  12. 12.

    Solve \(1 < 3 - 2x < 9\).

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    Subtract 3: \(-2 < -2x < 6\). Divide by \(-2\) (flip): \(1 > x > -3\), i.e. \(-3 < x < 1\). Answer: \((-3, 1)\).
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Compound Inequalities Practice Problems

The Compound Inequalities lesson explains intersection versus union and the three-part form; this set is 12 to practice on, each with a full solution in interval notation.

How to use this set

These problems are hand-written, not auto-generated. Solve each piece, then combine: overlap for “and”, both pieces for “or”. Write the answer in interval notation, then compare with the solution.

Common mistakes to watch for

Operating on only two of the three parts.

Turning an “or” into one interval. or stays as two pieces.

Forgetting to flip both symbols when dividing the three-part form by a negative.

Reporting a non-overlapping “and” as an interval instead of .

Where to go next

Each half is a Linear Inequality — drill those with the Linear Inequalities Practice Problems. The answer format is the Interval Notation Practice Problems. Compound inequalities power Absolute Value Inequalities. For the full method, see the Compound Inequalities lesson.

Frequently Asked Questions

What is the difference between 'and' and 'or' compounds?+

'And' (intersection) is satisfied only by values that make both inequalities true — the overlap. 'Or' (union) is satisfied by values that make at least one true — everything either piece covers.

How do I solve the three-part form a < x < b?+

Apply every operation to all three parts. : subtract 3 from each part, then divide each by 2.

When does an 'and' compound have no solution?+

When the two pieces don't overlap, like and . The answer is the empty set .

When does an 'or' compound cover all real numbers?+

When the two pieces together cover the line, like or . The answer is .

Do the symbols still flip when dividing by a negative?+

Yes. In the three-part form, dividing all three parts by a negative reverses both symbols; then rewrite so the smaller number is on the left.

How is the answer written?+

An 'and' result is one interval like . An 'or' result is a union: .

More practice problems

Linear Inequalities Practice Problems Interval Notation Practice Problems Absolute Value Inequalities Practice Problems