Chapter 4: Q. 48 (page 242)

Solve each inequality algebraically.
$\frac{{(2 - x)}^{3} (3 x - 2)}{x^{3} + 1} < 0$

Short Answer

Expert verified

Solution to the inequality $\frac{{(2 - x)}^{3} (3 x - 2)}{x^{3} + 1} < 0$ is $(- 1, \frac{2}{3}) \cup (2, \infty)$ .

Step by step solution

Step 1. Given information

We have been given an inequality $\frac{{(2 - x)}^{3} (3 x - 2)}{x^{3} + 1} < 0$ .

We have to solve this inequality algebraically.

Step 2. Determine the real numbers at which the expression f equals zero and at which the expression f is undefined.

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

${(2 - x)}^{3} = 0 2 - x = 0 x = 2 3 x - 2 = 0 3 x = 2 x = \frac{2}{3} x^{3} + 1 = 0 x^{3} = - 1 x = - 1$

Step 3. Form the intervals

Using the values of x found in previous step, we can divide the real numbers in the intervals:

$(- \infty, - 1) \cup (- 1, \frac{2}{3}) \cup (\frac{2}{3}, 2) \cup (2, \infty)$

Step 4. Select a number in each interval and evaluate f at the number

Create the following table:

Interval	$(- \infty, - 1)$	$(- 1, \frac{2}{3})$	$(\frac{2}{3}, 2)$	$(2, \infty)$
Number chosen	$- 2$	0	1	3
Value of f	$f (- 2) = \frac{512}{7}$	$f (0) = - 16$	$f (1) = \frac{1}{2}$	$f (3) = - \frac{1}{4}$
Conclusion	positive	negative	positive	negative

Step 5. Identify the interval

Since we want to know where f is negative, we conclude that $f (x) < 0$ in the interval localid="1646212202953" $(- 1, \frac{2}{3}) \cup (2, \infty)$ .

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Solve each inequality algebraically.
$\frac{{(2 - x)}^{3} (3 x - 2)}{x^{3} + 1} < 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the real numbers at which the expression f equals zero and at which the expression f is undefined.

Step 3. Form the intervals

Step 4. Select a number in each interval and evaluate f at the number

Step 5. Identify the interval

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Solve each inequality algebraically. (2-x)3(3x-2)x3+1<0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the real numbers at which the expression f equals zero and at which the expression f is undefined.

Step 3. Form the intervals

Step 4. Select a number in each interval and evaluate f at the number

Step 5. Identify the interval

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

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Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Solve each inequality algebraically.
$\frac{{(2 - x)}^{3} (3 x - 2)}{x^{3} + 1} < 0$