➕ Algebra 1 · Inequalities

Absolute Value Equations practice questions

Questions start easy and step up. Before you answer, rate how sure you are. It trains you to know what you really know.

Answer key with explanations

1. Which choice lists all the solutions of |x| = 14?

   Answer: x = 14 or x = −14. The numbers 14 units from 0 are 14 and −14, one on each side of 0. Check: |14| = 14 and |−14| = 14. Choosing only one of them forgets the second case.

2. Solve |x + 9| = 11.

   Answer: x = 2 or x = −20. Write x + 9 = 11 or x + 9 = −11. The first gives x = 2 and the second gives x = −20. Check: |2 + 9| = 11 and |−20 + 9| = |−11| = 11. The choice x = 11 or x = −11 ignores the +9 inside the bars: those solve |x| = 11.

3. Solve |3x − 3| = 12.

   Answer: x = 5 or x = −3. Write 3x − 3 = 12 or 3x − 3 = −12. The first gives 3x = 15, so x = 5. The second gives 3x = −9, so x = −3. Check: |3(5) − 3| = 12 and |3(−3) − 3| = |−12| = 12. The choice x = −5 comes from subtracting 3 instead of adding it in the second case. The choice x = 7 or x = −1 comes from dividing 3x and 12 by 3 but forgetting to divide the −3, which gives |x − 3| = 4.

4. Solve 5|x + 1| − 4 = 26.

   Answer: x = 5 or x = −7. Isolate the bars. Add 4: 5|x + 1| = 30. Divide by 5: |x + 1| = 6. Then x + 1 = 6 gives x = 5, and x + 1 = −6 gives x = −7. Check: 5|5 + 1| − 4 = 26 and 5|−7 + 1| − 4 = 5(6) − 4 = 26. The choice x = 29 or x = −31 forgets to divide by 5.

5. Solve |x/2 − 3| = 5.

   Answer: x = 16 or x = −4. Write x/2 − 3 = 5 or x/2 − 3 = −5. The first gives x/2 = 8, so x = 16. The second gives x/2 = −2, so x = −4. Check: |16/2 − 3| = 5 and |−4/2 − 3| = |−5| = 5. The choice x = 4 or x = −1 divides by 2 instead of multiplying by 2. The choice x = −16 comes from subtracting 3 instead of adding it in the second case.

6. Which equation has no solution?

   Answer: |4x + 1| + 5 = 1. Isolate the bars in each one. The first becomes |4x + 1| = −4, which has no solution. The second becomes |4x + 1| = 6, with two solutions. The third becomes |4x + 1| = 4, with two solutions. The fourth gives 4x + 1 = 0, so x = −1/4, one solution. A minus sign in the equation does not decide it: the number you end up with after isolating does.

7. A greenhouse thermostat is set to 24 degrees Celsius and is supposed to keep the temperature within 3 degrees of that. Which choice lists the two temperatures T that are exactly 3 degrees from the setting?

   Answer: 21 and 27 degrees Celsius. Exactly 3 degrees from 24 means |T − 24| = 3. Then T − 24 = 3 gives T = 27, and T − 24 = −3 gives T = 21. These are the two edges. Every temperature from 21 to 27 passes: 21 ≤ T ≤ 27. Choosing 27 only forgets the second case.

8. A trail marker sits at mile 11 on a straight trail. A rest stop at mile x is exactly 4 miles from that marker. Which equation describes the possible positions of the rest stop?

   Answer: |x − 11| = 4. |x − 11| is the distance between mile x and mile 11, so |x − 11| = 4. It gives x = 15 or x = 7. The equation |x + 11| = 4 puts the center at −11. The equation |x − 4| = 11 swaps the center and the distance: it gives mile 15 or mile −7, but mile 7 is the other correct position. The equation x − 11 = 4 allows only the stop that is past the marker, and misses mile 7.

9. Solve −3|x − 1| + 9 = −6.

   Answer: x = 6 or x = −4. Subtract 9: −3|x − 1| = −15. Divide by −3: |x − 1| = 5. The number is positive, so there are two cases: x − 1 = 5 gives x = 6, and x − 1 = −5 gives x = −4. Check: −3|6 − 1| + 9 = −6 and −3|−4 − 1| + 9 = −6. “No solution” comes from a sign slip: −15 ÷ (−3) is +5, not −5.

10. For which value of k does |x − 8| = k have exactly one solution?

    Answer: k = 0. One solution happens only when the distance is 0: |x − 8| = 0 gives x = 8. For k = 8 there are two solutions, x = 0 and x = 16. For k = 1 there are two solutions, x = 7 and x = 9. For k = −8 there is no solution, because a distance is never negative.