Question

How do you solve `(x-1)/(x+2)<0` using a sign chart?

Answer 1

The solution for the inequality is `-2 < x < 1`

Explanation:

As `(x-1)/(x+2)<0`, `x` cannot take value of `-2`.

From this we know that the product `(x-1)/(x+2)` is negative. It is apparent that sign of binomials `(x+2)` and `x-1` will change around the values `-2` and `1` respectively. In sign chart we divide the real number line using these values, i.e. below `-2`, between `-2` and `1` and above `1` and see how the sign of `(x-1)/(x+2)` changes.

Sign Chart

`color(white)(XXXXXXXXXXX)-2color(white)(XXXXX)1`

`(x+2)color(white)(XXX)-ive color(white)(XXXX)+ive color(white)(XXXX)+ive`

`(x-1)color(white)(XXX)-ive color(white)(XXXX)-ive color(white)(XXXX)+ive`

`(x-1)/(x+2)color(white)(XXX)+ive color(white)(XXX)-ive color(white)(XXXX)+ive`

It is observed that `(x-1)/(x+2) < 0` between `-2` and `1` i.e. `-2 < x < 1`, which is the solution for the inequality.