Answer:
x ∈ (-∞, -1) ∪ (1, ∞)
Step-by-step explanation:
To solve this problem we must factor the expression that is shown in the denominator of the inequality.
So, we have:
[tex]x ^ 2-1 = 0\\x ^ 2 = 1[/tex]
So the roots are:
[tex]x = 1\\x = -1[/tex]
Therefore we can write the expression in the following way:
[tex]x ^ 2-1 = (x-1)(x + 1)[/tex]
Now the expression is as follows:
[tex]\frac{(x-2) ^ 2}{(x-1) (x + 1)}\geq0[/tex]
Now we use the study of signs to solve this inequality.
We have 3 roots for the polynomials that make up the expression:
[tex]x = 1\\x = -1\\x = 2[/tex]
We know that the first two are not allowed because they make the denominator zero.
Observe the attached image.
Note that:
[tex](x-1)\geq0[/tex] when [tex]x\geq-1[/tex]
[tex](x + 1)\geq0[/tex] when [tex]x\geq1[/tex]
and
[tex](x-2) ^ 2[/tex] is always [tex]\geq0[/tex]
Finally after the study of signs we can reach the conclusion that:
x ∈ (-∞, -1) ∪ (1, 2] ∪ [2, ∞)
This is the same as
x ∈ (-∞, -1) ∪ (1, ∞)
