ACT Math · Elementary Algebra
Absolute value equations and inequalities
Solving |expression| = k and |expression| < k.
What's tested in this subtopic
Items test the absolute-value definition: |x| = k splits into x = k and x = −k; |x| < k splits into −k < x < k; |x| > k splits into x > k or x < −k.
Tactics that actually move your score
Always rewrite the absolute-value statement as the corresponding compound statement before solving. Check candidate solutions. The single biggest leverage point on most subtopics isn't learning more math — it's recognizing the test's preferred surface forms quickly enough that you don't burn 30 seconds re-reading the question. The first time you see a particular phrasing it might take you a full minute. The tenth time you see it, you should be reaching for your method before you've finished the sentence. Repetition is what builds that recognition. Fifteen problems in a row of the same shape is more useful than fifty mixed.
Practice questions (14)
- Easy What is the sum of all values of x that satisfy |x − 2| = 4?
- Easy What is the sum of all values of x that satisfy |x − 6| = 8?
- Medium What is the sum of all values of x that satisfy |x − 6| = 5?
- Medium What is the sum of all values of x that satisfy |x − 1| = 7?
- Medium What is the sum of all values of x that satisfy |x − 3| = 5?
- Hard What is the sum of all values of x that satisfy |x − 2| = 8?
- Easy What is the sum of all values of x that satisfy |x − 3| = 6?
- Easy What is the sum of all values of x that satisfy |x − 1| = 7?
- Medium What is the sum of all values of x that satisfy |x − 2| = 6?
- Medium What is the sum of all values of x that satisfy |x − 1| = 2?
- Medium What is the sum of all values of x that satisfy |x − 4| = 7?
- Hard What is the sum of all values of x that satisfy |x − 5| = 7?
- Easy What is the sum of all values of x that satisfy |x − 3| = 7?
- Easy What is the sum of all values of x that satisfy |x − 6| = 5?
How to drill
Work through the questions above untimed. After each one, read the worked solution from start to finish — even when you got it right. Note which solution method you used, and which method we used; if they differ, ask yourself which would have been faster on test day. Speed in ACT math comes from shortening your method-selection step, not from doing arithmetic faster. Most fast students are doing the same arithmetic everyone else is — they're just spending less time deciding what to do.
Once you can clear the easy and medium items in this subtopic at 90% accuracy, attempt a timed mini-set of ten hard items at 75 seconds each. If you finish in time and score 7+ correct, you've effectively mastered the subtopic for test purposes and can move on.