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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 11: Q. 365 (page 1069)

In the following exercises, solve each rational inequality and write the solution in interval notation.
$\frac{1}{2} - \frac{4}{x^{2}} 鈮� \frac{1}{x}$

Short Answer

Expert verified

Solution in interval notation is $x 鈭� [- 2, 0) 鈭� (0, 4] .$

Step by step solution

01

Step 1. Given Information

The rational inequality is given as $\frac{1}{2} - \frac{4}{x^{2}} 鈮� \frac{1}{x}$ .

02

Step 2. Inequality definition

On subtracting $\frac{1}{x}$ from both sides,

$\frac{1}{2} - \frac{4}{x^{2}} - \frac{1}{x} 鈮� \frac{1}{x} - \frac{1}{x} \frac{1}{2} - \frac{4}{x^{2}} - \frac{1}{x} 鈮� 0 \frac{x^{2} - 8 - 2 x}{2 x^{2}} 鈮� 0$

To define the inequality, the denominator should not be zero. One of the critical points are $x = 0 .$

03

Step 3. Polynomial function

The required condition for true inequality can be,

$x^{2} - 8 - 2 x 鈮� 0 x^{2} - 2 x - 8 鈮� 0 f (x) = x^{2} - 2 x - 8$

On factorizing the polynomial function using AC method.

${ax}^{2} + bx + c x^{2} - 2 x - 8 鈮� 0 x^{2} - 4 x + 2 x - 8 鈮� 0 x (x - 4) + 2 (x - 4) 鈮� 0 (x - 4) (x + 2) 鈮� 0$

04

Step 4. Critical points

The critical point for given polynomial function,

$f (x) = 0 (x - 4) (x + 2) = 0 x = 4, - 2$

Critical points are $x = 0, - 2, 4 .$

05

Step 5. Testing of critical points

The value of $\frac{x^{2} - 8 - 2 x}{2 x^{2}}$ can be tested,

$x = - 3 is \frac{(9) - 8 + 6}{18} = \frac{7}{18} x = - 1 is \frac{1 - 8 + 2}{2} = \frac{- 5}{2} x = 1 is \frac{1 - 8 - 2}{2} = \frac{- 9}{2} x = 5 is \frac{25 - 8 - 10}{50} = \frac{6}{50}$

Hence, $x 鈭� [- 2, 0) 鈭� (0, 4] .$

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Most popular questions from this chapter

Simplify

(a) ${(\frac{1}{2} y^{2})}^{3} {(\frac{2}{3} y)}^{2}$

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In the following exercises, factor each expression using any method $6 r^{2} + 30 r + 36$

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Solve: ${(9 x + 9)}^{\frac{1}{4}} - 2 = 1$

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