Answer:
x ∈ (-∞, -2) ∪ (2, ∞)
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 ^ 2-4 = 0\\x ^ 2 = 4[/tex]
So the roots are:
[tex]x = 2\\x = -2[/tex]
Therefore we can write the expression in the following way:
[tex]x ^ 2-4 = (x-2)(x + 2)[/tex]
Now the expression is as follows:
[tex]\frac{(x + 5) ^ 2}{(x-2)(x + 2)}\geq0[/tex]
Now we use the study of signs to solve this inequality.
We have 3 roots for the polynomials that compose the expression:
[tex]x = 2\\x = -2\\x = -5[/tex].
Observe the attached image.
We know that the first two roots are not allowed because they make zero the denominator, we also know that (x + 5) ^ 2 is always positive because it is squared, so it is not necessary to include the numerator in the study of signs.
Note that:
[tex](x-2)\geq 0[/tex] when [tex]x\geq2[/tex]
[tex](x + 2)\geq0[/tex] when [tex]x\geq-2[/tex]
Finally after the study of signs we can reach the conclusion that:
x ∈ (-∞, -2) ∪ (2, ∞)
