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Chapter 2: Problem 26

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

Short Answer

Expert verified
The solutions set of the inequality \(x^{2} \leq 2 x+2\) is [\(x_1, x_2\)], where \(x_1\) and \(x_2\) are roots of the equivalent equation \(x^{2} - 2x - 2 = 0\), found using the quadratic formula.

Step by step solution

01

Rearrange the equation

First, rearrange the inequality to one side which will look like this: \(x^{2} - 2x - 2 \leq 0\) which is a quadratic inequality.
02

Solve for the corresponding equation

Now, solve the corresponding equation, i.e., \(x^{2} - 2x - 2 = 0\). Use the quadratic formula \(x = \frac{-b 卤 \sqrt{b^{2} - 4ac} }{2a}\) to find the roots. In this equation, a is 1, b is -2 and c is -2. Then we find roots as \(x_1\) and \(x_2\).
03

Test intervals for the inequality

Test intervals defined by \(x_1\) and \(x_2\) for the inequality. Select a test point from each of interval \((-\infty, x_1)\), \((x_1, x_2)\), and \((x_2, \infty)\), substitute the test point into the inequality, if the inequality is satisfied then that interval is a part of the solution set.
04

Express solution in interval notation and graph on a number line

Express the solution set in interval notation and plot these intervals on real number line.

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Most popular questions from this chapter

Exercises \(61-63\) will help you prepare for the material covered in the first section of the next chapter. Use point plotting to graph \(f(x)=2^{x}\). Begin by setting up a partial table of coordinates, selecting integers from -3 to \(3,\) inclusive, for \(x .\) Because \(y=0\) is a horizontal asymptote, your graph should approach, but never touch, the negative portion of the \(x\) -axis.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\)-intercept.

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. $$\frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}}$$

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

a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\). b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\).
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