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Problem 434

AMC 12 late, AIME early

Algebra Difficulty 4.8 Find the answer

Solve the inequality $\frac{x+2}{3 x+1} \leq \frac{x-2}{2 x-1}$ .

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Official solution

Solution: $\frac{x+2}{3 x+1} \leq \frac{x-2}{2 x-1} ; \frac{x+2}{3 x+1}-\frac{x-2}{2 x-1} \leq 0, \frac{(x+2)(2 x-1)-(x-2)(3 x-1)}{(3 x+1)(2 x-1)} \leq 0$ , $\frac{2 x^{2}+3 x-2-3 x^{2}+5 x+2}{(3 x+1)(2 x-1)} \leq 0, \frac{-8 x^{2}+8 x}{(3 x+1)(2 x-1)} \leq 0, \frac{-x(x-8)}{(3 x+1)(2 x-1)} \leq 0$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.