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Snezhnost [94]
3 years ago
8

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

Mathematics
1 answer:
Digiron [165]3 years ago
6 0

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:

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:

\frac{(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, ∞)

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