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**hooke** Solve this inequality

$\displaystyle |2x-1|+|x+2|\geq 4x$

so i have to for 3 different cases , $\displaystyle x\geq 1/2$ , $\displaystyle -2\leq x<1/2$ , $\displaystyle x<-2 $

For $\displaystyle x<-2$ , $\displaystyle -(2x-1)-(x+2)\geq 4x$

$\displaystyle x\leq -\frac{1}{7}$

For $\displaystyle x\geq \frac{1}{2}$ ,

$\displaystyle 2x-1+x+2\geq 4x$

$\displaystyle x\leq 1$

For $\displaystyle -2 \leq x < \frac{1}{2}$ ,

$\displaystyle -(2x-1)+x+2\geq 4x$

$\displaystyle x\leq \frac{3}{5}$

after combining , the solution would be $\displaystyle x\leq -\frac{1}{7}$

AM i correct ? but the answer given is $\displaystyle x\leq 1$ ??