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Chapter 2: Problem 31

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$x(3-x)(x-5) \leq 0$$

Short Answer

Expert verified
The solution set of the inequality \(x(3-x)(x-5) \leq 0\) in interval notation is [0, 3) \(\cup\) (5, \(\infty\)).

Step by step solution

01

Identify the Critical Points

Set each factor of the inequality to zero to identify the critical points. This can be done by setting \(x = 0\), \(3 - x = 0\) and \(x - 5 = 0\). Accordingly, the critical points are \(x = 0\), \(x = 3\), and \(x = 5\).
02

Create a Number Line

Next, create a number line that includes the critical points \(0\), \(3\), and \(5\). The critical points separate the number line into four intervals which are \(-\infty, 0\), \(0, 3\), \(3, 5\), and \(5, \infty \).
03

Test Intervals

Test an x-value from each interval in the original inequality. In the first interval (\(-\infty, 0\)), an x-value may be \(-1\). Substituting it yields a positive value, so this interval is not part of the solution set. For the second interval (\(0, 3\)), an x-value can be \(1\). Substituting it yields a negative value, thus, this interval is part of the solution set. For the third interval (\(3, 5\)), \(x = 4\) can be chosen which results in a positive value. Hence, this interval is not part of the solution set. In the fourth interval (\(5, \infty\)), substituting \(x = 6\) yields a negative value, which makes it part of the solution set.
04

Express in Interval Notation

The solution set in interval notation is \([0, 3) \cup (5, \infty)\), taking into consideration that \(0\) and \(5\) are part of the solution set as the inequality allows equality.

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Most popular questions from this chapter

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