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Chapter 2: Problem 70

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

Short Answer

Expert verified
The solution to the inequality \(\frac{x^{2}-3 x+2}{x^{2}-2 x-3} > 0\) is \((- \infty, -1) \cup (1, 2) \cup (3, \infty)\). Be sure to note that \(x = -1\) and \(x = 3\) are undefined.

Step by step solution

01

Find and Plot Critical Points

To find the critical points, set the numerator \(x^{2}-3 x+2=0\) and the denominator \(x^{2}-2 x-3=0\) equal to zero separately, and solve for \(x\). So, \(x = 1, 2\) from the numerator and \(x = -1, 3\) from the denominator. Plot these points on the number line.
02

Test Intervals for Sign Changes

Next, pick one test point from each interval to check the sign of the fraction. For instance, you can choose \(x = -2, 0, 1.5, 2.5 \) and \(4\) in the interval \((- \infty, -1)\), \((-1, 1)\), \((1, 2)\), \((2, 3)\) and \((3, \infty)\) respectively. Substitute each \(x\) into the fraction and determine if it's positive or negative.
03

Determine Solution Set Based on Inequality Sign

Because the inequality is '>', we need to find intervals where the fraction is positive. If the fraction is positive for all \(x\) in that interval, then the interval is part of the solution set. When you are finished, make sure to clearly mark your solution set on the number line.
04

Exclude Undefined Points

Since fractions are undefined when the denominator is zero, we need to exclude the points that make denominator zero from the solution set. In this case, those points are \(x = -1\) and \(x = 3\).

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Most popular questions from this chapter

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+4}{x}$$

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