Question

How do you solve `(3-x)/(x+5)<=0` using a sign chart?

Answer 1

`(3-x)/(x+5) <= 0` when either `x < -5` or `x >= 3`, which is the solution for the inequality. Note that at `x=-5`, it is not defined.

Explanation:

The sign of `(3-x)/(x+5)` depends on the signs of binomials `(3-x)` and `x+5`, which should change around the values `-5` and `3` respectively. In sign chart we divide the real number line using these values, i.e. below `-5`, between `-5` and `3` and above `3` and see how the sign of `(3-x)/(x+5)` changes.

Sign Chart

`color(white)(XXXXXXXXXXX)-5color(white)(XXXXX)3`

`(3-x)color(white)(XXX)+ive color(white)(XXXX)+ive color(white)(XXXX)-ive`

`(x+5)color(white)(XXX)-ive color(white)(XXXX)+ive color(white)(XXXX)+ive`

`(3-x)/(x+5)color(white)(XXX)-ive color(white)(XXXX)+ive color(white)(XXXX)-ive`

It is observed that `(3-x)/(x+5) <= 0` when either `x < -5` or `x >= 3`, which is the solution for the inequality. Note that at `x=-5`, it is not defined.
graph{(3-x)/(x+5) [-20, 20, -10, 10]}