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Chapter 2: Problem 25

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

Short Answer

Expert verified
The solution to the inequality \(x^{2} \leq 4x -2\) in interval notation is \((2 - \sqrt{2}, 2 + \sqrt{2})\).

Step by step solution

01

Rearrange the inequality

First, rearrange the inequality to get it to the form \(f(x) \leq 0\). As we have \(x^{2} \leq 4x - 2\), we subtract \(4x - 2\) from both sides to get \(x^{2} - 4x +2 \leq 0\).
02

Find the roots of the equivalent equation

Set \(x^{2} - 4x +2 = 0\) to find the zeros of the inequality. We can use the quadratic formula for this which is \(x = [4 \pm \sqrt{(4)^{2}-4*1*2}]/2*1\), giving us \(x = 2 \pm \sqrt{2}\), so we have two roots \(x = 2 + \sqrt{2}\) and \(x = 2 - \sqrt{2}\). These points will be used to divide the number line into intervals.
03

Identify the intervals

Using the roots from step 2, divide the number line into intervals. The intervals are \((-鈭�, 2 - \sqrt{2})\), \((2 - \sqrt{2}, 2 + \sqrt{2})\), and \((2 + \sqrt{2}, 鈭�)\). Now, pick a test point from each interval and substitute it in the inequality \(x^{2} - 4x +2 \leq 0\) to check the sign of the inequality.
04

Test the intervals

Choosing \(x = 0\) (in the first interval), \(x = 2\) (in the second interval), and \(x = 3\) (in the third interval). Upon substituting these values we find that the first interval gives a positive result, the second interval gives a negative result and the third interval gives a positive result. Therefore, the inequality is satisfied in the interval \((2 - \sqrt{2}, 2 + \sqrt{2})\).
05

Express the solution set in interval notation

The solution is the interval where the inequality holds true, so it is \((2 - \sqrt{2}, 2 + \sqrt{2})\).

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Simplify: \(\frac{x+1}{x+3}-2\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I began the solution of the rational inequality \(\frac{x+1}{x+3} \geq 2\) by setting both \(x+1\) and \(x+3\) equal to zero.

A company is planning to manufacture mountain bikes. The fixed monthly cost will be \(\$ 100,000\) and it will cost \(\$ 100\) to produce each bicycle. a. Write the cost function, \(C,\) of producing \(x\) mountain bikes. b. Write the average cost function, \(\bar{C},\) of producing \(x\) mountain bikes. c. Find and interpret \(\bar{C}(500), \bar{C}(1000), \bar{C}(2000),\) and \(\bar{C}(4000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(\bar{C} ?\) Describe what this means in practical terms.

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+1}{x}$$

What is a polynomial inequality?
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