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Algebra Difficulty 3.0 AMC 10/12 Find the answer Canada

For each value of $x$ , $f(x)$ is defined to be the minimum value of the three numbers $2x+2$ , $\dfrac{1}{2}x+1$ and $-\dfrac{3}{4}x+7$ . What is the maximum value of $f(x)$ ?

(A) $\dfrac{2}{3}$ (B) $2$ (C) $\dfrac{17}{5}$ (D) $\dfrac{62}{11}$ (E) $7$

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.