Chapter 11: Q. 365 (page 1069)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

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

Step 2. Inequality definition

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

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

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

Step 3. Polynomial function

The required condition for true inequality can be,

$x^{2} - 8 - 2 x \leq 0 x^{2} - 2 x - 8 \leq 0 f (x) = x^{2} - 2 x - 8$

On factorizing the polynomial function using AC method.

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

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 $x = 0, - 2, 4 .$

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, $x \in [- 2, 0) \cup (0, 4] .$

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Inequality definition

Step 3. Polynomial function

Step 4. Critical points

Step 5. Testing of critical points

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In the following exercises, solve each rational inequality and write the solution in interval notation.12-4x2≤1x

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Inequality definition

Step 3. Polynomial function

Step 4. Critical points

Step 5. Testing of critical points

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Geometry

Pure Maths

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

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