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Math Help - inequality

  1. #1
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    inequality

    Solve this inequality

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

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

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

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

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

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

    x\leq 1

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

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

    x\leq \frac{3}{5}

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

    AM i correct ? but the answer given is x\leq 1 ??
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  2. #2
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    Quote Originally Posted by hooke View Post
    Solve this inequality

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

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

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

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

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

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

    x\leq 1

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

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

    x\leq \frac{3}{5}

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

    AM i correct ? but the answer given is x\leq 1 ??
    Did you consider just checking it? If x= 0, for example, |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 x\le 1.

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