Free GMAT Problem Solving Practice Question

Underline two things in the stem: the absolute-value bars and the phrase 'sum of all values.' Because the variable appears inside the bars and on the right side, the equation splits into two cases, and every value found must be checked for validity.

Write the two cases. Case 1, the inside is non-negative (2v − 9 ≥ 0, so v ≥ 4.5): 2v − 9 = v + 3, which gives v = 12. Check: 12 ≥ 4.5, and the right side v + 3 = 15 is positive, so this root is valid. Case 2, the inside is negative (v < 4.5): take the opposite sign, −(2v − 9) = v + 3, so −2v + 9 = v + 3, giving 3v = 6 and v = 2. Check: 2 < 4.5, and the right side v + 3 = 5 is positive, so this root is valid too.

Both cases produce a valid value, so the complete solution set is {12, 2}. The question asks for the sum, so the answer is 12 + 2 = 14, choice D. (As a cross-check, squaring both sides gives the quadratic v² − 14v + 24 = 0, whose two roots sum to 14, the same total.)

Why the wrong choices tempt: choice C (12) is the classic absolute-value mistake of solving only the positive-inside case and forgetting that the bars also allow the negative-inside case. Choice A (2) is the mirror error of keeping only the other root. Choice B (10) comes from finding both roots but subtracting instead of adding, as if the second case should be counted negatively. Choice E (24) comes from multiplying the two roots instead of summing them, a misread of the word 'sum.'