Question

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


`((x+5)^(2) )/(x^(2)-4 ) \geq 0`

Answer 1

x ∈ (-∞, -2) ∪ (2, ∞)

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:

`((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, ∞)