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Chapter 3: Problem 12

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. $$ 9 x^{2}+3 x-2 \geq 0 $$

Short Answer

Expert verified
The solution of the polynomial inequality \(9x^{2}+3x-2 \geq 0\) in interval notation is \((-鈭�, -0.24] \cup [0.91, 鈭�)\)

Step by step solution

01

Find the roots

Using the quadratic formula \(-b卤\sqrt{b^{2}-4ac}/2a\) to solve for \(x\), we have \(x = -3卤\sqrt{3^{2}-4*9*(-2)}/18\). Therefore, we get two roots \(x_1\) and \(x_2\) which are approximates -0.24 and 0.91.
02

Identify the intervals

With roots \(-0.24\) and \(0.91\), we now have three intervals: \((-鈭�, -0.24)\), \((-0.24, 0.91)\), and \((0.91, 鈭�)\). These intervals are created by the zeros of the function.
03

Test the intervals

Now, we need to test a value in each interval against the inequality. For example, choose x=-1 for interval (-鈭�, -0.24), x=0 for interval (-0.24, 0.91), and x=1 for interval (0.91, 鈭�), and substitute each in the inequality to see if true.
04

Write the solution in interval notation

The solution set in interval notation will be the intervals that satisfy the inequality. From the analysis above, we see that for every \(x\) from the interval \((-鈭�, -0.24]\) and for every \(x\) from the interval \([0.91, 鈭�)\) the inequality holds true.

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Most popular questions from this chapter

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. $$ \frac{x-5}{10 x-2}+\frac{x^{2}-10 x+25}{25 x^{2}-1} $$

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. $$f(x)=-3 x^{3}(x-1)^{2}(x+3)$$

What are the zeros of a polynomial function and how are they found?

In Exercises 100鈥�103, determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed \(f(x)=(x+2)^{3}(x-4)^{2},\) and the graph touched the \(x\) -axis and turned around at \(-2\)

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.
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