Question

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


`((x-2)^(2) )/(x^(2)-1 ) \geq 0`

Answer 1

x ∈ (-∞, -1) ∪ (1, ∞)

Explanation:

To solve this problem we must factor the expression that is shown in the denominator of the inequality.

So, we have:

`x ^ 2-1 = 0\\x ^ 2 = 1`

So the roots are:

`x = 1\\x = -1`

Therefore we can write the expression in the following way:

`x ^ 2-1 = (x-1)(x + 1)`

Now the expression is as follows:

`((x-2) ^ 2)/((x-1) (x + 1))\geq0`

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

We have 3 roots for the polynomials that make up the expression:

`x = 1\\x = -1\\x = 2`

We know that the first two are not allowed because they make the denominator zero.

Observe the attached image.

Note that:

`(x-1)\geq0` when
`x\geq-1`

`(x + 1)\geq0` when
`x\geq1`

and

`(x-2) ^ 2` is always
`\geq0`

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, ∞)