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Chapter 2: Problem 69

Solve each inequality and graph the solution set on a real number line. $$\frac{x^{2}-x-2}{x^{2}-4 x+3}>0$$

Short Answer

Expert verified
The solution to the given rational inequality is \(-\infty<x<-1,1< x <2,3<x<+\infty\).

Step by step solution

01

Factorize the Given Rational Inequality

First, you need to factorize the numerator and the denominator of your rational inequality. Therefore, \( \frac{x^{2}-x-2}{x^{2}-4 x+3} \) becomes \( \frac{(x-2)(x+1)}{(x-1)(x-3)} > 0 \).
02

Find the Critical Points

The critical points are values of x that make the numerator or the denominator equal to zero. They partition the number line into several intervals. Set each factor in the numerator and denominator equal to zero and solve for x. Here, the critical points are x=2, x=-1, x=1, and x=3.
03

Test the Intervals

The number line is divided into the five intervals by the critical points: \(-\infty, -1\), \(-1, 1\), \(1, 2\), \(2, 3\), and \(3, +\infty\). You will now select one number from each of these intervals and evaluate it in the simplified inequality. If the value of the inequality is positive, the interval is part of the solution set. If it's negative, it's not.
04

List the Solution Intervals

After evaluating the test numbers and determining which intervals are positive and which are negative, the solution intervals for this rational inequality equation are: \(-\infty, -1\) (not included), \(1, 2\) (not included), \(3, +\infty\) (not included). You'll write these as \(-\infty<x<-1,1< x <2,3<x<+\infty\).

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