Question

How do you solve the inequality `x^2-4x<=5`?

Answer 1

The answer is `x in [ -1, 5 ]`

Explanation:

Let's rewrite the inequality

`x^2-4x-5<=0`
Let's factorise the LHS

`(x+1)(x-5)<=0`

Let `f(x)=(x+1)(x-5)`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-1` `color(white)(aaaaa)` `5` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+1` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-5` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)<=0` when `x in [ -1, 5 ]`