solve the inequality. x^2 - x over x^2 +5x is less than or equal to 0?

Question

express the answer in terms of intervals if possible. may need inf and cup symbols.

Answer 1

Step 1: determine boundary points by setting the top = 0, solve for x, and setting the bottom = 0, solve for x.

x = 0, 1 from the top
x = 0, -5 from the bottom

Step 2: test the intervals (- infinity, -5), (-5, 0), (0, 1) & (1, infinity)

(- infinity, -5) is False
(-5, 0) is True
(0, 1) is True
(1, infinity) is False

*if I didn't make an arithmetic mistake. I have a cold, so double check these, ok??

We include x = 1 in the solution, since that gives us the "or equal to." We do NOT include x = -5 or 0, since they give us division by zero.

If I did ok, then your answer is (-5, 0) U (0, 1]

Answer 2

Factor: x(x-1)/[x(x+5)] <=0
Then (x-1)/(x+5) <= 0
x can't be greater than 1, for expression is positive
x can't equal 1, since it becomes not determined
x can't be -5 since denom becomes 0
x can't be less than -5 since expression is again positive.
So -5