Question

How do you solve the inequality `((x-3)(x-4))/((x-5)(x-6)^2)<=0`?

Answer 1

The answer is `x in ] -oo,3] uu [4, 5[`

Explanation:

Let `f(x)=((x-3)(x-4))/((x-5)(x-6)^2)`

The domain of `f(x)` is `D_f(x)=RR-{5,6}`

And `(x-6)^2>=0`

To solve this inequality, we need to establish a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `3` `color(white)(aaaa)` `4` `color(white)(aaaa)` `5` `color(white)(aaaa)` `6` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x-3` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaa)` `+` `color(white)(aa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaa)` `+` `color(white)(aa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-5` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaa)` `-` `color(white)(aa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaa)` `-` `color(white)(aa)` `+` `color(white)(aaa)` `+`

Therefore, `f(x)<=0`

when, `x in ] -oo,3] uu [4, 5[`
graph{((x-3)(x-4))/((x-5)(x-6)^2) [-1.575, 6.22, -2.234, 1.666]}