# Solving inequality with modulus

• Jun 2nd 2011, 11:39 PM
Punch
Solving inequality with modulus
Solve the inequality $\displaystyle (x+2)(x-3)>|2x-1|$

$\displaystyle (x+2)(x-3)>2x-1$ or $\displaystyle (x+2)(x-3)<1-2x$

Solving the first inequality: $\displaystyle x^2-x-6>2x-1$

$\displaystyle x^2-3x-5>0$

$\displaystyle x<-1.19$ or $\displaystyle x>4.19$

Solving the second inequality: $\displaystyle x^2-x-6<1-2x$

$\displaystyle x^2+x-7<0$

$\displaystyle -3.19<x<2.19$

so, $\displaystyle x<-1.19$ or $\displaystyle x>4.19$ or $\displaystyle -3.19<x<2.19$

But the answer is $\displaystyle x<-3.19$ or $\displaystyle x>4.19$

• Jun 3rd 2011, 12:15 AM
FernandoRevilla
Quote:

Originally Posted by Punch
Solve the inequality $\displaystyle (x+2)(x-3)>|2x-1|$

$\displaystyle (x+2)(x-3)>2x-1$ or $\displaystyle (x+2)(x-3)<1-2x$

It should be:

$\displaystyle (x+2)(x-3)>|2x-1|\Leftrightarrow$

$\displaystyle \begin{Bmatrix} (x+2)(x-3)>2x-1 & \mbox{ if }& x\geq 1/2\\(x+2)(x-3)>1-2x & \mbox{if}& x<1/2\end{matrix}$
• Jun 3rd 2011, 12:35 AM
Punch
but how would that change my answer?
• Jun 3rd 2011, 02:37 AM
HallsofIvy
You found x= -1.19 as a solution to (x+2)(x-3)> 2x-1 but |2x-1|= 2x-1 only if 2x-1>= 0 or x>= 1/2, as FernandoRevilla said. -1.19 is NOT greater than 1/2.