Question

How do you solve the inequality `2x^2+25x+63<=0`?

Answer 1

The answer is `x in [-9, -7/2]`

Explanation:

Let's factorise the LHS

`2x^2+25x+63<=0`

`(2x+7)(x+9)<=0`

Let `f(x)=(2x+7)(x+9)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-9` `color(white)(aaaa)` `-7/2` `color(white)(aaaa)` `-oo`

`color(white)(aaaa)` `x+9` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `2x+7` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)<=0`, when `x in [-9, -7/2]`