Absolute Value — ACT Math Guide

Q: What You Need to Know

• Absolute value measures distance from zero, so it's always non-negative • |x| = x when x ≥ 0, and |x| = -x when x 0): solution is -a

Absolute value ACT questions appear regularly on the math section and test your understanding of distance from zero. The absolute value of a number is always positive (or zero), representing how far that number sits from zero on the number line. You'll typically see 2-3 absolute value problems among the 60 questions in 60 minutes on the ACT math section. With clear rules and consistent practice, absolute value becomes one of the more predictable topics you can master.

What You Need to Know

Absolute value measures distance from zero, so it's always non-negative

|x| = x when x ≥ 0, and |x| = -x when x < 0

|a| = |b| means a = b OR a = -b (two solutions possible)

Absolute value equations can have 0, 1, or 2 solutions

For |x| < a (where a > 0): solution is -a < x < a

For |x| > a (where a > 0): solution is x < -a OR x > a

📐 KEY FORMULA: |x| = distance from x to 0 on the number line

⏱️ ACT TIME TIP: Absolute value problems are usually in the first half of the section — solve quickly to save time for harder geometry questions later

How to Solve Absolute Value on the ACT

Example Question 1 — Easy/Medium Difficulty

What is the value of |3 - 7| + |-4|?

A) -8

B) -4

C) 4

D) 6

E) 8

Solution:

Step 1: Calculate |3 - 7| = |-4| = 4

Step 2: Calculate |-4| = 4

Step 3: Add the results: 4 + 4 = 8

✅Answer: E — The absolute value removes negative signs, giving us 4 + 4 = 8.

Example Question 2 — Hard Difficulty

For what values of x does |2x - 6| = 10?

A) x = -2 only

B) x = 8 only

C) x = -2 and x = 8

D) x = 2 and x = -8

E) No solution exists

Solution:

Step 1: Set up two equations: 2x - 6 = 10 OR 2x - 6 = -10

Step 2: Solve first equation: 2x = 16, so x = 8

Step 3: Solve second equation: 2x = -4, so x = -2

✅Answer: C — Absolute value equations typically yield two solutions when the right side is positive.

Common ACT Math Mistakes to Avoid

❌Mistake: Forgetting that |x| = x when x is already positive

✅Fix: Remember absolute value only changes negative numbers to positive

❌Mistake: Solving |x| = -5 and getting confused about "no solution"

✅Fix: Absolute value cannot equal a negative number — these have no solution

❌Mistake: Missing the second solution in equations like |x - 3| = 7

✅Fix: Always consider both x - 3 = 7 AND x - 3 = -7

❌Mistake: Confusing inequality directions when |x| > a vs |x| < a

✅Fix: Draw a number line — |x| < a means x is between -a and a

Practice Question — Try It Yourself

If |x + 2| ≤ 5, which of the following represents all possible values of x?

A) x ≤ 3

B) x ≥ -7

C) -7 ≤ x ≤ 3

D) x ≤ -7 or x ≥ 3

E) -3 ≤ x ≤ 7

Show Answer

Answer: C — The inequality |x + 2| ≤ 5 means -5 ≤ x + 2 ≤ 5, which gives us -7 ≤ x ≤ 3.

Key Takeaways for the ACT

Absolute value always produces non-negative results — negative answers are usually wrong

Most ACT math absolute value equations have exactly two solutions

Use your calculator freely since ACT allows calculators throughout the math section

Absolute value inequalities create ranges (≤) or unions of ranges (≥)

When stuck, plug the answer choices back into the original equation — all five options make this strategy viable