Question

How do you solve `x/(x-2)> -1/(x+3)` using a sign chart?

Answer 1

The solution is `x in (-oo,-4.45] uu(-3, 0.45] uu (2, +oo)`

Explanation:

Simplify the inequality, we cannot do crossing over.

`x/(x-2) > -1/(x+3)`

`x/(x-2)+1/(x+3)>0`

`(x(x+3)+(x-2))/((x-2)(x+3))>0`

`(x^2+3x+x-2)/((x-2)(x+3)) >0`

`(x^2+4x-2)/((x-2)(x+3)) >0`

The roots of the numerator

`x^2+4x-2=0`

are

`x=(-4+-sqrt(16-4(1)(-2)))/(2)`

`=-2+-sqrt6`

`x_1=-2-sqrt6=-4.45`

`x_2=-2+sqrt6=0.45`

Let

`f(x)=((x-x_1)(x-x_2))/((x-2)(x+3))`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `x_1` `color(white)(aaaa)` `-3` `color(white)(aaaaa)` `x_2` `color(white)(aaaaa)` `2` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x-x_1` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)`##

`color(white)(aaaa)` `x-3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aa)` `||` `color(white)(a)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-x_2` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aa)` ###color(white)(aa)#`-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aa)` ###color(white)(aa)#`-` `color(white)(aaaa)` `-` `color(white)(aa)` `||` `color(white)(a)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aa)` `||` `color(white)(a)` `+` `color(white)(aaaa)` `-` `color(white)(aa)` `||` `color(white)(a)` `+`

Therefore,

`f(x) >0` when `x in (-oo,-4.45] uu(-3, 0.45] uu (2, +oo)`

graph{(x^2+4x-2)/((x-2)(x+3)) [-10, 10, -5, 5]}