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Chapter 3: Problem 97

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I began the solution of the rational inequality \(\frac{x+1}{x+3} \geq 2\) by setting both \(x+1\) and \(x+3\) equal to zero.

Short Answer

Expert verified
No, the approach is incorrect. To solve a rational inequality, neither the numerator nor the denominator should be separately set to zero. Instead, the entire rational expression needs to be set to zero to solve for the critical points.

Step by step solution

01

Understand the Approach

In the provided solution approach, it is suggested to set both the numerator (\(x+1\)) and the denominator (\(x+3\)) of the rational inequality to zero. However, this is not a correct approach. In order to solve a rational inequality, neither the numerator nor the denominator should be simply set to zero. Instead, the entire rational expression is set to zero to find the critical points, and these points are used to test for the inequalities.
02

Correct Approach for Solving Rational Inequality

To solve the inequality \(\frac{x+1}{x+3} \geq 2\), first set the entire expression equal to zero. It will be: \( \frac{x+1}{x+3} - 2 = 0\). That simplifies to: \( \frac{x+1-2x-6}{x+3} = 0\) or \(\frac{-x-5}{x+3}=0\) . Solving for \(x\), first find when the numerator is zero and then when the denominator is zero. Those are the critical points, which divide the number line into intervals that can be tested for the inequalities.
03

Conclusion

Setting the numerator and denominator of the inequality to zero - as was originally done - does not yield the correct solution to the inequality. Instead, the entire rational inequality should be set equal to zero to isolate x, and then the resulting x-values are used to find the solution intervals of the inequality.

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