Question

How do you solve `x/(x+6)>=0`?

Answer 1

`x >= 0 and x < -6`

Explanation:

Before we begin, we have an implicit restriction of `x != -6`

For the case where `x/(x + 6) = 0`, we merely set the numerator equal to 0 obtain the answer `x = 0`

For the case where `x/(x + 6) > 0`, we have two cases, one where signs of the numerator and denominator are both positive and one where the of the numerator and denominator are both negative.

For both the both positive case, we have two inequalities:

`x > 0 and x + 6 > 0`

Simplify the second inequality:

`x > 0 and x > -6`

Please observe that there are numbers from -6 to 0 that would cause the numerator to become negative, therefore, we must use `x > 0`

For the both negative case, we have two inequalities:

`x < 0 and x + 6 < 0`

Again, we simplify:

`x < 0 and x < -6`

Again, the numbers from 0 to -6 would cause a conflict, therefore, we use `x < -6`

Combining the 3 answers we obtain the two ranges:

`x < - 6 and x >= 0`