Question

How do you solve `x^2<=9` using a sign chart?

Answer 1

The answer is `x in [-3,3]`

Explanation:

Let's rewrite the equation

`x^2<=9`

`x^2-9<=0`

`(x+3)(x-3)<=0`

Let `f(x)=(x+3)(x-3)`

Now, we can build the sign chart

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

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

`color(white)(aaaa)` `x-3` `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 [-3,3]`