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Chapter 2: Problem 34

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$(5-x)^{2}\left(x-\frac{13}{2}\right)<0$$

Short Answer

Expert verified
The solution to the given polynomial inequality is \( (-\infty, 5) \cup (5, \frac{13}{2}) \)

Step by step solution

01

Factorize the Polynomial Inequality

The given polynomial inequality \[(5-x)^{2}\left(x-\frac{13}{2}\right)<0\] is already factorized.
02

Find the Boundaries of the Solution Set

Set each factor equal to zero and solve for \(x\). This gives \[5-x=0 \quad \implies x = 5\]and\[x-\frac{13}{2}=0 \quad \implies x=\frac{13}{2}\]
03

Test the Different Regions Defined by the Boundaries

The boundaries divide the real line into three regions. Select a test point from each region and substitute it into the inequality. For \(x<5\) use \(x=0\), for \(5\frac{13}{2}\) use \(x=7\). Evaluating the inequality at each test point, we find that the inequality is satisfied when \(x<5\) or \(5<x<\frac{13}{2}\)
04

Express the Solution Set in Interval Notation

Using interval notation, the solution set can be written as \( (-\infty, 5) \cup (5, \frac{13}{2}) \)

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Most popular questions from this chapter

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