Question

How do you solve `(x-3)^2<=1` using a sign chart?

Answer 1

The answer is `x in [2,4]`

Explanation:

We need

`a^2-b^2=(a+b)(a-b)`

Let's rewrite the inequality

`(x-3)^2<=1`

`(x-3)^2-1<=0`

`(x-3-1)(x-3+1)<=0`

`(x-4)(x-2)<=0`

Let `f(x)=(x-4)(x-2)`

Now, we build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaa)` `-oo` `color(white)(aaaa)` `2` `color(white)(aaaaa)` `4` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `x-2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaaaa)` `-` `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 [2,4]`