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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
Which of the following gives all values of n that satisfy |n + 6| = 10?
n = 4 only
n = −16 or n = 4
n = −4 or n = 16
n = −4 or n = 4
Answer: B. n = −16 or n = 4
Split into n + 6 = 10 (n = 4) and n + 6 = −10 (n = −16). A keeps only the positive case; C subtracts 6 with the wrong sign in both cases; D negates the answer 4 instead of setting the whole expression n + 6 equal to −10.
Question 2
How many real solutions does the equation |x + 8| + 5 = 2 have?
0
1
2
Infinitely many
Answer: A. 0
Subtract 5: |x + 8| = −3. An absolute value is never negative, so there are 0 solutions. Answering 2 solves x + 8 = ±3 as if the right side were positive; 1 treats it like a single linear equation; infinitely many has no basis here.
Question 3
Which of the following equations has no real solution?
|2x − 3| + 4 = 1
|x| − 6 = −2
−|x + 1| = −5
|5 − x| = 0
Answer: A. |2x − 3| + 4 = 1
Isolating in A gives |2x − 3| = −3, which is impossible. B becomes |x| = 4 (two solutions); in C the negatives cancel to |x + 1| = 5 (two solutions); D has exactly one solution, x = 5, since an absolute value can equal 0.
Question 4
For which of the following equations is x = −2 a solution?
|x − 5| = 3
|x + 2| = 2
|x + 5| = 3
|2x + 1| = 5
Answer: C. |x + 5| = 3
Substitute x = −2: |−2 + 5| = |3| = 3, so C works. In A, |−2 − 5| = 7, not 3 (its solutions are 2 and 8, the sign trap). In B, |0| = 0, not 2. In D, |−4 + 1| = 3, not 5 (its solutions are 2 and −3).
Question 5
On a number line, the point with coordinate x is the same distance from −7 as it is from 3, so |x + 7| = |x − 3|. What is the value of x?
−5
−2
2
5
Answer: B. −2
The equal case x + 7 = x − 3 is impossible; the opposite case x + 7 = −(x − 3) gives 2x = −4, so x = −2, the midpoint of −7 and 3. −5 is half of −7 − 3 rather than half of −7 + 3; 2 flips the sign; 5 is the distance from the midpoint to either point.
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