Question

How do you solve `x/(x-3)<=-8/(x-6)` using a sign chart?

Answer 1

The solution is `x in [-6,3[ uu [4,6 [`

Explanation:

We cannot do crossing over

`x/(x-3)<=-8/(x-6)`

We rearrange and factorise the inequality

`x/(x-3)+8/(x-6)<=0`

`(x(x-6)+8(x-3))/((x-3)(x-6))<=0`

`(x^2-6x+8x-24)/((x-3)(x-6))<=0`

`(x^2+2x-24)/((x-3)(x-6))<=0`

`((x-4)(x+6))/((x-3)(x-6))<=0`

Let `f(x)=((x-4)(x+6))/((x-3)(x-6))`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-6` `color(white)(aaaaaaa)` `3` `color(white)(aaaaa)` `4` `color(white)(aaaaaaa)` `6` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+6` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aa)` `+` `color(white)(aa)` `+` `color(white)(aaaa)` `||` `color(white)(aa)` `+`

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

`color(white)(aaaa)` `x-4` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `-` `color(white)(aa)` `+` `color(white)(aaaa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `x-6` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `-` `color(white)(aa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `+`

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

Therefore,

`f(x)<=0` when `x in [-6,3[ uu [4,6 [`