SAT Math: Absolute Value Equations Practice Questions
Treat absolute value as distance, split equations into valid cases, and distinguish two solutions from one or none. Try each original question before checking its worked explanation. This focused practice is free and requires no account.
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1. What is the sum of the real solutions of |2x - 5| = 9?
Explanation. Set 2x - 5 equal to 9 or -9. The solutions are 7 and -2, whose sum is 5. Both give a distance of 9 from zero inside the absolute value.
2. What is the larger solution of |x + 4| = 7?
Explanation. The two cases are x + 4 = 7 and x + 4 = -7, giving x = 3 and x = -11. The larger is 3.
3. How many real solutions does |3x + 1| = -4 have?
Explanation. An absolute value is nonnegative for every real input. It cannot equal -4, so the equation has zero real solutions.
4. What value of x satisfies |5x - 20| = 0?
Explanation. An absolute value is zero exactly when its input is zero. Solve 5x - 20 = 0 to obtain x = 4. This equation has one solution rather than two distinct cases.
5. A point on a number line is 6 units from 2. What is the product of its two possible coordinates?
Explanation. The coordinates are 2 + 6 = 8 and 2 - 6 = -4, which solve |x - 2| = 6. Their product is 8(-4) = -32.
6. What is the smaller real solution of 2|x - 1| + 3 = 11?
Explanation. Subtract 3 and divide by 2: |x - 1| = 4. Thus x - 1 = 4 or -4, giving x = 5 or -3. The smaller solution is -3.
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FAQ
When does |u| = a have two cases?
When a is positive, solve u = a and u = -a. When a is zero, solve u = 0. When a is negative, there is no real solution.
How does absolute value represent distance?
The expression |x - c| is the distance between x and c on the number line. A positive specified distance usually gives one point on each side of c.