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Chapter 3: Problem 31

Solve each polynomial inequality in Exercises \(1-42\) 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 in interval notation is \([0, 3]\).

Step by step solution

01

Find Critical Points

Solve the equation \(x(3-x)(x-5) = 0\) for x. This yields the points 0, 3, and 5. These are the critical points.
02

Setup the Number Line

Place these critical points on a number line, which creates four intervals. Those intervals are \(-\infty, 0\), \(0, 3\), \(3, 5\), and \(5, \infty\).
03

Testing Each Interval

Since the inequality includes equals to 0, include the critical points in the solution. Test a number in each interval to see if it satisfies the inequality. For \(-\infty, 0\), let's pick \(-1\), the inequality simplifies to a negative, thus this interval is not a part of the solution. For \(0, 3\), pick \(1\), which simplifies to a positive, hence this interval is part of the solution. Do the same for \(3, 5\) and \(5, \infty\), pick 4 and 6 respectively. The inequality simplifies to a negative for \(4\) and positive for \(6\). Therefore, the intervals \(3, 5\) and \(5, \infty\) are not included in the solution set.
04

Write the Solution in Interval Notation

Combine the solution intervals to write the solution in interval notation: \([0, 3] \)

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Most popular questions from this chapter

In Exercises \(98-99,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \([\mathrm{ZOOMOUT}]\) feature to show that \(f\) and \(g\) have identical end behavior. $$f(x)=-x^{4}+2 x^{3}-6 x, \quad g(x)=-x^{4}$$

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In Exercises 104鈥�107, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a function with origin symmetry can rise to the left and rise to the right.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the ZOOM OUT feature to show that f and g have identical end behavior. \(f(x)=-x^{4}+2 x^{3}-6 x, \quad g(x)=-x^{4}\)

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