Q:

Solving Rational Inequalities and use sign diagram to sketch the graph. Image attached for better understanding.[tex]\frac{(x-2)^{2} }{x^{2}-1 } \geq 0[/tex]

Accepted Solution

A:
Answer:x ∈ (-∞, -1) βˆͺ (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:[tex]x ^ 2-1 = 0\\x ^ 2 = 1[/tex]So the roots are:[tex]x = 1\\x = -1[/tex]Therefore we can write the expression in the following way:[tex]x ^ 2-1 = (x-1)(x + 1)[/tex]Now the expression is as follows:[tex]\frac{(x-2) ^ 2}{(x-1) (x + 1)}\geq0[/tex]Now we use the study of signs to solve this inequality.We have 3 roots for the polynomials that make up the expression:[tex]x = 1\\x = -1\\x = 2[/tex]We know that the first two are not allowed because they make the denominator zero.Observe the attached image.Note that:[tex](x-1)\geq0[/tex] when [tex]x\geq-1[/tex][tex](x + 1)\geq0[/tex] when [tex]x\geq1[/tex]and[tex](x-2) ^ 2[/tex] is always [tex]\geq0[/tex]Finally after the study of signs we can reach the conclusion that:x ∈ (-∞, -1) βˆͺ (1, 2] βˆͺ [2, ∞)This is the same asx ∈ (-∞, -1) βˆͺ (1, ∞)