Chapter 4: Q. 46 (page 242)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information

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

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

$x = 0 x^{2} + 1 = 0 T h e r e i s n o s u c h r e a l x . x - 2 = 0 x = 2 x - 1 = 0 x = 1 x + 1 = 0 x = - 1$

Step 3. Form the intervals

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

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, 2) \cup (2, \infty)$

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

Create the following table:

Interval	$(- \infty, - 1)$	$(- 1, 0)$	$(0, 1)$	$(1, 2)$	$(2, \infty)$
Number chosen	$- 2$	$- 0.5$	$0.5$	$1.5$	4
Value of f	$f (- 2) = \frac{40}{3}$	$f (- 0.5) = - 2.08$	$f (0.5) = 1.25$	$f (1.5) = - 1.95$	$f (4) = \frac{136}{15}$
Conclusion	positive	negative	positive	negative	positive

Step 5. Identify the interval

Since we want to know where f is positive or zero, we conclude that $f (x) \geq 0$ in the interval $(- \infty, - 1) \cup [0, 1) \cup [2, \infty)$ .

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

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. x(x2+1)(x-2)(x-1)(x+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

Mechanics Maths

Geometry

Pure Maths

Statistics

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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