How do I solve this? Been stuck on it forever!
$\displaystyle |x+2|\geq{x\over{x+2}}$
You need to be careful if you're squaring both sides of an inequality:
$\displaystyle 0.5 > - 3$
$\displaystyle \Rightarrow 0.25 > 9$? I don't think so.
If $\displaystyle x = -1.5$, then $\displaystyle |x+2| = 0.5$ and $\displaystyle \frac{x}{x+2} = -3 \dots$ ???
I think instead, you'll need to look at the signs of $\displaystyle (x+2)$ and $\displaystyle |x+2|$ in the cases
- $\displaystyle x>-2$
- $\displaystyle x<-2$
and consider each one as a separate inequality. Have you tried this approach?
Grandad
1. The domain is $\displaystyle d = \mathbb{R} \setminus \{-2\}$
2. $\displaystyle |x+2| = \left\{ \begin{array}{l} x+2,\ x >-2 \\ -(x+2),\ x < -2\end{array}\right.$
3. $\displaystyle x+2\geq\dfrac x{x+2}\ ,\ x > -2$
$\displaystyle (x+2)^2-x\geq 0~\implies~\left(x+\frac32\right)^2+\dfrac74\geq 0$
is true for all x because both summands at the LHS are positive.
Therefore $\displaystyle x>-2$
4. $\displaystyle -(x+2) \geq \dfrac x{x+2}\ ,\ x < -2$
$\displaystyle -x^2-5x-4 \bold{{\color{red}\leq}} 0~\implies~(x+1)(x+4) \bold{{\color{red}\geq}} 0~\implies~x \geq -1\wedge x\geq -4~\vee~ x \leq -1\wedge x \leq -4$
Therefore $\displaystyle x \leq -4$
Thanks o_O!
EDIT: I've removed my mistake. Corrections in red!
Small correction in red.
Reason being:
$\displaystyle \begin{aligned} -(x+2) & \geq \frac{x}{x+2} \\ -(x+2)^2 & \ {\color{red} \leq} \ x \qquad \text{Since (x+2) is negative, inequality sign changes} \\ -x^2 - 4x - 4 & \leq x \\ -x^2 - 5x - 4 & \leq 0 \\ & \ \ \vdots \end{aligned} $