# Help with absolute value inequalitie

I am having trouble finding what the answer to this inequalitie is: $$|x+4|x + |x+2| > -2$$

first i did this:

|x+4|=\left\{ \begin{align} x+4 & \text{ , if }x\geq -4 \\ -(x+4) & \text{ , if }x <-4 \end{align} \right\}

|x+2|=\left\{ \begin{align} x+2 & \text{ , if }x\geq -2 \\ -(x+2) & \text{ , if }x <-2 \end{align} \right\}

Then i got this:

$$1. x<-4:$$

$$-(x+4)x + (-(x+2)) > -2$$

$$2. -4$$\le$$x<-2:$$

$$(x+4)x + (-(x+2)) > -2$$

$$3. x$$\geq$$-2:$$

$$(x+4)x+x+2 > -2$$

After i calculated top 3 inequalities i got this answers:

$$1. x$$^2$$+ 5x < 0$$

$$2. x$$^2$$+ 3x > 0$$

$$3. x$$^2$$+5x+4 > 0$$

I think i calculated everything correctly, but now i do not know how to find the answer to this absolute value inequalitie.

Thank you for your help.

## 1 Answer

Guide:

For the first case, we need $$x<-4$$ and $$x^2+5x<0$$

that is $$x<-4$$ and $$x(x+5) <0$$ which is equivalent to

$$x<-4$$ and $$-5. which can be summarized as $$-5.

Do the same thing for the other cases as well.

Remark: $$x(x+5)<0$$ is equivalent to $$-5 can be seen from its graph: 