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ACT · Math · Preparing for Higher Math: Algebra
Absolute value equations is part of Preparing for Higher Math: Algebra in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include absolute value equation with two solutions, isolate the absolute value first, no solution case and distance on a number line.
Common traps: reporting only the positive case; not changing the sign of the whole expression in the second case; distributing the negative incorrectly.
Question 1
What is the sum of the solutions of |4 − 3x| = 10?
−8/3
8/3
14/3
20/3
Answer: B. 8/3
4 − 3x = 10 gives x = −2; 4 − 3x = −10 gives x = 14/3. The sum is 14/3 − 2 = 8/3. −8/3 has the sign reversed (each solution negated); 14/3 is only one solution; 20/3 adds the absolute values of the solutions.
Question 2
The two solutions of |x − 2.5| = 7 are plotted on a number line. What is the distance between the two points?
4.5
7
9.5
14
Answer: D. 14
The solutions are 2.5 + 7 = 9.5 and 2.5 − 7 = −4.5, so the distance between them is 9.5 − (−4.5) = 14, twice the 7. 7 is the distance from each point to 2.5, not between the points; 9.5 is the greater solution; 4.5 drops the sign of −4.5.
Question 3
What is the sum of all real numbers x that satisfy |x − 4| = 9?
−5
4
8
13
Answer: C. 8
The points 9 units from 4 are x = 13 and x = −5. Their sum is 13 + (−5) = 8, which is twice the center 4. Choice A (−5) gives only one of the two solutions, 4 − 9. Choice B (4) is the center of the two points, not their sum. Choice D (13) gives only one of the two solutions, 4 + 9.
Question 4
What is the sum of the squares of all solutions to 3|2x − 3| + 4 = 31?
3
9
18
45
Answer: D. 45
|2x − 3| = 9, so 2x − 3 = ±9 and x = 6 or x = −3. The squares sum to 36 + 9 = 45. Choice A (3) gives the sum of the solutions. Choice B (9) squares the sum instead of summing squares. Choice C (18) gives the absolute value of the product.
Question 5
The equation |ax − 6| = 10, where a is a positive constant, has x = 4 as one of its solutions. What is the other solution?
−4
−1
1
4
Answer: B. −1
With x = 4, |4a − 6| = 10 gives 4a − 6 = 10, so a = 4 (the case 4a − 6 = −10 gives a = −1, which is not positive). Then 4x − 6 = −10 gives x = −1. −4 assumes the solutions are opposites; 1 slips a sign in 4x = −4; 4 repeats the given solution.
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