1. ## inequality

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 $\displaystyle x\leq 1$ ??

2. Originally Posted by 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 $\displaystyle x\leq 1$ ??
Did you consider just checking it? If x= 0, for example, $\displaystyle |2x-1|+ |x+2|= |-1|+ |2|= 3\ge 0= 4x$ so "x< -1/7" is clearly not correct.

Your analysis for -2< x< 1/2 gave x< 3/5 which is certainly true of all x< 1/2< 3/5 so YOUR calculations say that any number less than 1/2 satisfies the inequality.

Your analysis for x> 1/2 give x< 1 so all numbers between 1/2 and 1 also satisfy the inequality. What YOU did shows that the inequality is satisfied for all $\displaystyle x\le 1$.

Your only mistake is the last step where you say "after combining , the solution would be $\displaystyle x\leq -\frac{1}{7}$" which does not follow at all.