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Chapter 3: Problem 19

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. $$ x^{2}-4 x \geq 0 $$

Short Answer

Expert verified
The solution of the inequality \(x^{2}-4 x \geq 0\) in interval notation is \([0, 4]\)

Step by step solution

01

Set the Inequality Equation to Zero

Rearrange the inequality to have it in the form \(f(x) \geq 0\). This simplifies the polynomial inequality to \(x^2 - 4x = 0\)
02

Find the Roots of the Equation

Solving the equation \(x^2 - 4x = 0\) gives the roots of the equation. This can be solved by factoring out x from the equation: \(x(x - 4) = 0\). Setting this equal to zero gives \(x = 0, 4\). These are the roots of the equation
03

Test the Intervals on the Real Number Line

Divide the real number line into three intervals using the roots: \((-鈭�, 0)\), \((0, 4)\), and \((4, +鈭�)\). Test each interval with a number from the interval in the original inequality equation to find where the equation holds. For interval \((-鈭�, 0)\) with test number -1, the inequality does not hold. For interval \((0, 4)\) with test number 2, the inequality holds. Lastly for interval \((4, +鈭�)\) with test number 5, the inequality does not hold.
04

Express the Solution in Interval Notation

Based on the test for each interval, the solution to the polynomial inequality in interval notation, including the roots, is \([0, 4]\)

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Most popular questions from this chapter

Describe a strategy for graphing a polynomial function. In your description, mention intercepts, the polynomial鈥檚 degree, and turning points.

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

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

Use a graphing utility to graph \(y=\frac{1}{x^{\prime}}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)
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