Question

How do you solve `1/2x^2<=x+12` using a sign chart?

Answer 1

The solution is `x in [-4 ,6]`

Explanation:

Let's rearrange the inequality

`1/2x^2<= x+12`

`1/2x^2- x-12<=0`

`x^2- 2x-24 <=0`

Factorising, we get

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

Let, `f(x)=(x+4)(x-6)`

We can build the sign chart

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

`color(white)(aaaa)` `x+4` `color(white)(aaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-6` `color(white)(aaaaaa)` `-` `color(white)(aa)` `color(white)(aaaa)` `-` `color(white)(a)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `+` `color(white)(aa)` `0` `color(white)(aaa)` `-` `color(white)(a)` `0` `color(white)(aa)` `+`

Therefore,

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

graph{1/2x^2-x-12 [-16.22, 15.8, -12.55, 3.47]}