Chapter 7: Q. 366 (page 732)

In the following exercises, solve each rational inequality and write the solution in interval notation.
$\frac{1}{2} - \frac{3}{2 x^{2}} \geq \frac{1}{x}$

Short Answer

Expert verified

Solution for rational inequality is $x \in (- \infty, 1] \cup [3, \infty) .$

Step by step solution

Step 1. Given Information

Given rational inequality is $\frac{1}{2} - \frac{3}{2 x^{2}} \geq \frac{1}{x}$ .

Step 2. Definition of inequality

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

$\frac{1}{2} - \frac{3}{2 x^{2}} - \frac{1}{x} \geq \frac{1}{x} - \frac{1}{x} \frac{1}{2} - \frac{3}{2 x^{2}} - \frac{1}{x} \geq 0 \frac{x^{2} - 3 - 2 x}{2 x^{2}} \geq 0$

For the definition of inequality, denominator should not be zero. Thus, one of the critical points is $x = 0 .$

Step 3. Required condition

To true the above inequality, the condition can be stated as,

$x^{2} - 3 - 2 x \geq 0 f (x) = x^{2} - 2 x - 3$

On factorizing the polynomial using AC method.

${ax}^{2} + bx + c x^{2} - 3 - 2 x \geq 0 x^{2} + x - 3 x - 3 \geq 0 x (x + 1) - 3 (x + 1) \geq 0 (x + 1) (x + 3) \geq 0$

Step 4. Critical points

To find the critical points,

$f (x) = 0 x^{2} - 2 x - 3 = 0 (x + 1) (x - 3) = 0 x = - 1, 3$

Thus, the critical points are $x = 0, - 1, 3 .$

Step 5. Testing of critical points

To test these critical points, value of the $\frac{x^{2} - 3 - 2 x}{2 x^{2}}$ at different points.

$x = - 2 is \frac{4 - 3 + 4}{8} = \frac{5}{8} x = 0.5 is \frac{0.25 - 3 - 1}{0.5} = - 7.5 x = 2 is \frac{4 - 3 - 4}{8} = \frac{- 3}{8} x = 4 is \frac{16 - 3 - 8}{32} = \frac{5}{32}$

This satisfies the condition of $x \in (- \infty, - 1] \cup [3, \infty) .$

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In the following exercises, solve each rational inequality and write the solution in interval notation.
$\frac{1}{2} - \frac{3}{2 x^{2}} \geq \frac{1}{x}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Definition of inequality

Step 3. Required condition

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.12-32x2≥1x

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Definition of inequality

Step 3. Required condition

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

Discrete Mathematics

Statistics

Mechanics Maths

Applied Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

In the following exercises, solve each rational inequality and write the solution in interval notation.
$\frac{1}{2} - \frac{3}{2 x^{2}} \geq \frac{1}{x}$