Question

How do you solve `(x+5)/(x-2)>0`?

Answer 1

`x in (-oo, -5) uu (2, oo)`

Explanation:

`(x+5)/(x-2) > 0`

This inequality will hold when both the numerator and denominator are positive or both negative.

The values of `x` at which the numerator or denominator change sign are `x=-5` and `x=2`.

So the intervals we need to consider are:

`(-oo, -5)`, `(-5, 2)` and `(2, oo)`

When `x in (-oo, -5)` both the numerator and denominator are negative, so the quotient is positive as required.

When `x in (-5, 2)` the numerator is positive but the denominator is negative, so the quotient is negative.

When `x in (2, oo)` both the numerator and denominator are positive, so the quotient is positive.

So the solution space is:

`(-oo, -5) uu (2, oo)`

graph{(y(x-2)-(x+5))(x+0.0001y-2)(y-1+0.001x) = 0 [-10.63, 9.37, -4.44, 5.56]}