Question

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

Answer 1

The answer is `x in ] -oo,1 ] uu [3, +oo[`

Explanation:

We rewrite the equation as

`1-1/(x-2)^2>=0`

We do some simplifications

`((x-2)^2-1)/(x-2)^2>=0`

`((x-2-1)(x-2+1))/(x-2)^2>=0`

`((x-3)(x-1))/(x-2)^2>=0`

Let `f(x)=((x-3)(x-1))/(x-2)^2`

The domain of `f(x)` is `D_f(x)=RR-{2}`

The denominator is `>0, AA x in D_f(x)`

We can do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `1` `color(white)(aaaa)` `2` `color(white)(aaaa)` `3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x-1` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `∣∣` `color(white)(aa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `∣∣` `color(white)(aa)` `-` `color(white)(aaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `-` `∣∣` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)>=0`, when `x in ] -oo,1 ] uu [3, +oo[`

graph{(y-((x-1)(x-3))/(x-2)^2)=0 [-8.89, 8.89, -4.444, 4.445]}