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Chapter 2: Problem 42

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$x^{3} \leq 4 x^{2}$$

Short Answer

Expert verified
\((-\infty, 0] \cup [4, \infty)\)

Step by step solution

01

Re-arrange the inequality

Subtract \(4x^2\) from both sides to get it to the standard format for solving inequalities. It should look like: \(x^{3} - 4x^{2} \leq 0\)
02

Find the roots

To find the roots, set the inequality to be equal to zero and fix x. That results in \(x^{3} - 4x^{2} = 0\). Factoring out the common term \(x^2\), we have \(x^2(x-4) = 0\), which gives two values: \(x = 0\) and \(x = 4\).
03

Define the intervals

The roots divide the number line into three intervals: \(-\infty < x < 0\), \(0 < x < 4\), and \(4 < x < \infty\).
04

Test each interval

Pick a test point in each interval and substitute it into the original inequality. If it fulfills the inequality, the whole interval is part of the solution. The test points could be, for example, \(x=-1\), \(x=2\), and \(x=5\). Doing so results in the intervals \(-\infty < x \leq 0\) and \(4 \leq x < \infty\) as the solutions to the inequality.
05

Write the solution in interval notation

Finally, the solution can be written in interval notation as \((-\infty, 0] \cup [4, \infty)\)

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Most popular questions from this chapter

Describe how to graph a rational function.

a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\). b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\).

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{2 x+7}{x+3}$$

Find the horizontal asymptote, if there is one, of the graph of rational function. $$h(x)=\frac{15 x^{3}}{3 x^{2}+1}$$

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)
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