Inequalities and Absolute Value Equations Quiz

Grade 9 · Math · Quiz

Test your mastery of solving and interpreting inequalities, compound inequalities, and absolute value problems.

Questions

1. Solve: 4x − 3 ≤ 13

   1. x ≤ 4
   2. x ≥ 4
   3. x ≤ 2.5
   4. x ≥ −4
   Show answer

   Answer: x ≤ 4

   4x − 3 ≤ 13 → 4x ≤ 16 → x ≤ 4.

2. Solve: −6 < 2x + 4 ≤ 10

   1. −5 < x ≤ 3
   2. −1 < x ≤ 7
   3. −5 ≤ x < 3
   4. −1 ≤ x < 7
   Show answer

   Answer: −5 < x ≤ 3

   Subtract 4: −10 < 2x ≤ 6. Divide by 2: −5 < x ≤ 3.

3. Solve |x − 3| = 10.

   1. x = 7 or x = −7
   2. x = 13 or x = −7
   3. x = 7 or x = −13
   4. x = 13 only
   Show answer

   Answer: x = 13 or x = −7

   x − 3 = 10 → x = 13, or x − 3 = −10 → x = −7.

4. Solve |4x − 8| ≤ 12.

   1. −1 ≤ x ≤ 5
   2. x ≤ −1 or x ≥ 5
   3. −5 ≤ x ≤ 1
   4. x ≤ −5 or x ≥ 1
   Show answer

   Answer: −1 ≤ x ≤ 5

   |4x − 8| ≤ 12 means −12 ≤ 4x − 8 ≤ 12. Add 8: −4 ≤ 4x ≤ 20. Divide by 4: −1 ≤ x ≤ 5.

5. Which inequality has no solution?

   1. |x + 5| = 0
   2. |x − 2| > −1
   3. |x + 3| < −4
   4. 2|x| > 0
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   Answer: |x + 3| < −4

   |x + 3| < −4 has no solution because absolute value is always ≥ 0 and can never be less than a negative number.

6. Solve: −2(x + 3) > 4x − 6

   1. x > 0
   2. x < 0
   3. x > −6
   4. x < −6
   Show answer

   Answer: x < 0

   Expand: −2x − 6 > 4x − 6. Add 2x: −6 > 6x − 6. Add 6: 0 > 6x. Divide by 6: x < 0.

7. The absolute value equation |2x + 6| = 0 has exactly one solution.

   Show answer

   Answer: True

   |2x + 6| = 0 means 2x + 6 = 0, so x = −3. There is exactly one solution.

8. Solve |x| > 8. Write the solution as x < −8 or x > __.

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   Answer: 8

   |x| > 8 splits into x > 8 or x < −8 because any number more than 8 units from zero satisfies the condition.

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