# Thread: Absolute Value Inequality

1. ## Absolute Value Inequality

solve the inequality

$\displaystyle \left| 5x+4\right| >6$

2. Originally Posted by OzzMan
solve the inequality

$\displaystyle \left| 5x+4\right| >6$
$\displaystyle |5x+4| = \left\{\begin{array}{lcr}5x+4 &if& x\geq -\frac45\\ -(5x+4)&if &x < -\frac45\end{array}\right.$

That means you have 2 inequalities:

$\displaystyle \begin{array}{lcr}5x+4>6 &if& x\geq -\frac45\\ -(5x+4)>6&if &x < -\frac45\end{array}$

$\displaystyle \begin{array}{lcr}x> \frac25 &if& x\geq -\frac45\\ x<-2&if &x < -\frac45\end{array}$

The solution of this inequality is: $\displaystyle x < -2~\vee~x > \frac25$

3. I see my error. Thanks. Just one question though. How come you wrote $\displaystyle 5x+4\quad if\quad x\ge -\frac{4}{5}\quad instead\; of\quad x>-\frac{4}{5}$

4. Originally Posted by OzzMan
I see my error. Thanks. Just one question though. How come you wrote $\displaystyle 5x+4\quad if\quad x\ge -\frac{4}{5}\quad instead\; of\quad x>-\frac{4}{5}$
The definiton of an absolute value is:

$\displaystyle |x| = \left\{\begin{array}{lcr}x & if & x \geq 0\\-(x) & if & x <0 \end{array}\right.$

I only tried to use the definition correctly