## mathmath333 one year ago Find the range of $$x$$.

1. mathmath333

\large \color{black}{\begin{align} \dfrac{x-3}{x+5}\leq 0\hspace{.33em}\\~\\ \end{align}}

2. idku

x has to be greater than -5 and smaller than or equal to 3 so that either top or bottom is negative (or that it is 0).

3. idku

$$\Large -5<x\le 3$$

4. anonymous

numerator and denominator must be of opposite sign

5. geerky42

We can see that $$x\neq-5$$. Multiply both sides by $$x+5$$, from here, split it into two cases. $x-3\le0\quad \text{if }~~x+5>0$ $x-3\ge0\quad \text{if }~~x+5<0$ For first case, you have $$x\le3$$ if $$x>-5$$, so range is $$-5<x\le3$$ Second case is impossible.

6. imqwerty

x belongs to (-5,3]

7. idku

$$\LARGE x\in (-5,3]$$

8. idku

hate notations, and love them

9. idku

it feels good sometimes that I am capable of helping - not just asking questions. tnx, and yw