how do i write this in interval form??
(x+2)/(x-5)≤0
You should first note that $\displaystyle x \neq 5$.
$\displaystyle \frac{x + 2}{x - 5} \leq 0$
$\displaystyle 1 + \frac{7}{x - 5} \leq 0$
$\displaystyle \frac{7}{x - 5} \leq 1$
Now we will have two alternatives, depending on whether the denominator is positive or negative...
Case 1: $\displaystyle x - 5$ is positive, which means $\displaystyle x > 5$.
$\displaystyle 7 \leq x - 5$
$\displaystyle 14 \leq x$
$\displaystyle x \geq 14$.
Case 2: $\displaystyle x - 5$ is negative, which means $\displaystyle x < 5$.
$\displaystyle 7 \geq x - 5$
$\displaystyle 14 \geq x$
$\displaystyle x \leq 14$.
It's pretty obvious that the ONLY $\displaystyle x$ which can not satisfy this equation are $\displaystyle x = 5$.
So the domain is $\displaystyle x < 5 \cup x > 5$.
Alternatively, it can be written $\displaystyle x \neq 5$ or $\displaystyle (-\infty , 5) \cup (5, \infty)$.
please be specific. are we talking about writing the domain in interval notation or the solution to the inequality in interval notation?
I suspect the latter. In which case set the numerator and denominator to zero. You get x = -2 or x = 5 (now note that x cannot be 5, because the denominator cannot in fact be zero, we just do this to figure out the signs). Now you can draw a number line and plot x = -2 and x = 5 on it. Then plug in numbers in each of the three intervals that you have. you will find that the middle interval is the only one that works. the resulting solution is $\displaystyle [-2, 5)$
A fraction is negative if and only if numerator and denominator are of different signs. That is:
1) $\displaystyle x+2\ge 0$ and x- 5< 0
which lead to $\displaystyle x\ge -2$ and x< 5: [-2, 5)
or
2) $\displaystyle x+2\le 0$ and x- 5> 0
which lead to $\displaystyle x\le -2$ and x> 5 which are impossible together.
HallsofIvy's answer is indeed correct. To demonstrate why $\displaystyle (-\infty,5)\cup(5,\infty)$ is wrong, try substituting $\displaystyle x=6$ into the inequality and see what you get. Actually, any $\displaystyle x>5$ gives a positive result, in contradiction to this answer.
However, IF the original question was "what is the domain of the function $\displaystyle \frac{x+2}{x- 5}$, THEN the answer is $\displaystyle (-\infty, 5)\cup(5, \infty)$. Prove It said that in his first response!
However, I don't see what the "<=" has to do with it then. You find domains of functions, not inequalities.