1962 AMC 12 Problem 29

Attempt Problem 29 of the 1962 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1962 AMC 12 solutions, or check the answer key.

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29.

Which of the following sets of xx-values satisfy the inequality 2x2+x<6?2x^2+x\lt6?
−2<x<32-2\lt x\lt\dfrac32
x>32x\gt\dfrac32 or x<−2x\lt-2
x<32x\lt\dfrac32
32<x<2\dfrac32\lt x\lt2
x<−2x\lt-2
Answer: A
Concepts:inequalityfactoringzero product property
Small hint:
Move all terms to one side and factor the quadratic
Big hint:
A product of two linear factors is negative between its roots
Solution:
The inequality is 2x2+x−6<0.2x^2+x-6\lt0. Factoring gives (2x−3)(x+2)<0.(2x-3)(x+2)\lt0. The product is negative between its roots, so −2<x<32.-2\lt x\lt\frac{3}{2}.
Thus, the correct answer is A.
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