Question

Solving Rational Inequalities and use sign diagram to sketch the graph. Image attached for better understanding.

Answer 1

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:

$x ^ 2-4 = 0\\x ^ 2 = 4$

So the roots are:

$x = 2\\x = -2$

Therefore we can write the expression in the following way:

$x ^ 2-4 = (x-2)(x + 2)$

Now the expression is as follows:

$\frac{(x + 5) ^ 2}{(x-2)(x + 2)}\geq0$

Now we use the study of signs to solve this inequality.

We have 3 roots for the polynomials that compose the expression:

$x = 2\\x = -2\\x = -5$.

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:

$(x-2)\geq 0$ when $x\geq2$

$(x + 2)\geq0$ when $x\geq-2$

Finally after the study of signs we can reach the conclusion that:

x ∈ (-∞, -2) ∪ (2, ∞)