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Chapter 3: Problem 33

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. $$ (2-x)^{2}\left(x-\frac{7}{2}\right)<0 $$

Short Answer

Expert verified
The solution to the inequality in interval notation is \((2,\frac{7}{2})\).

Step by step solution

01

Factor the Inequality

Factor the given inequality \( (2-x)^{2}\left(x-\frac{7}{2}\right)<0 \). It's already factored so we proceed to the next step.
02

Find the Critical Points

The critical points are obtained by setting each factor equal to zero. Thus, for \(2-x = 0\), \(x = 2\) and for \(x-\frac{7}{2} = 0\), \(x = \frac{7}{2}\). Therefore, the critical points are \(2\) and \(\frac{7}{2}\).
03

Plot the Critical Points on a Number Line

Place the critical points on a number line. The number line will be divided into three intervals by the points \(2\) and \(\frac{7}{2}\).
04

Test the Intervals

Choose a number from each interval to evaluate the inequality. If the number makes the inequality true, then the interval is part of the solution set. For \(x<2\), we could choose \(1\); for \(2\frac{7}{2}\), we could choose \(4\). Using these test points produces \(9>0\), \(-4<0\), and \(9>0\) respectively.
05

Express the Solution in Interval Notation

The interval where the inequality holds true is when \(2<x<\frac{7}{2}\). In interval notation, this is expressed as \((2, \frac{7}{2})\).

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Most popular questions from this chapter

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. There is more than one third-degree polynomial function with the same three \(x\) -intercepts.

What is a rational function?

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=3 x^{2}-x^{3}$$

Why is a third-degree polynomial function with a negative leading coefficient not appropriate for modeling nonnegative real-world phenomena over a long period of time?

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{x}{2 x+6}-\frac{9}{x^{2}-9} $$
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