Question

Find the range of \(x\).

Answer 1

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

Answer 2

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).

Answer 3

\(\Large -5<x\le 3 \)

Answer 4

numerator and denominator must be of opposite sign

Answer 5

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.

Answer 6

x belongs to (-5,3]

Answer 7

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

Answer 8

hate notations, and love them

Answer 9

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

tnx, and yw