Chapter 4: Q. 54 (page 242)

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

Short Answer

Expert verified

Solution set and the interval is:
${x | x < - 2 o r 1 \leq x \leq \infty} (- \infty, - 2) \cup [1, \infty)$

Step by step solution

Step 1. Change the inequality to equal to zero.

Change the inequality equal to zero to make it easier and then get the value of x
First arrange the inequality,

$\begin{aligned} \frac{x - 1}{x + 2} \geq - 2 \\ \frac{x - 1}{x + 2} + 2 \geq 0 \\ \frac{x - 1}{x + 2} + 2 \cdot \frac{x + 2}{x + 2} \geq 0 \\ \frac{x - 1 + 2 x + 4}{x + 2} \geq 0 \\ \frac{3 x - 3}{x + 2} \geq 0 \end{aligned}$

Make it equal to zero.

$\begin{array}{rr} \frac{3 x - 3}{x + 2} = 0 \\ (3 x - 3) (x + 2) = 0 \\ 3 x - 3 = 0; x + 2 = 0 \\ 3 x = 3; x = - 2 \\ x = \frac{3}{3} \\ x = 1 \end{array}$

Step 2. Form the intervals

From the obtained values of x, we can form the interval,

So the interval we have is:

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

Step 3. Form the table

Since, $f (x) \geq 0$ , so is positive or zero.

check a number in each interval and evaluate the function to see whether it is satisfying the function.

Interval	Number chosen	Resultant
$(- \infty, - 2)$	-3	$12 \geq 0$
$(- 2, 1)$	0	$- \frac{3}{2} \geq 0$
$(1, \infty)$	2	$\frac{3}{4} \geq 0$

Step 4. Solution set and interval

Since, $12 \geq 0, \frac{3}{4} \geq 0$ satisfy $f$ . Thus,

role="math" localid="1646208744522" ${x | x < - 2 o r 1 \leq x \leq \infty} (- \infty, - 2) \cup [1, \infty)$

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

Short Answer

Step by step solution

Step 1. Change the inequality to equal to zero.

Step 2. Form the intervals

Step 3. Form the table

Step 4. Solution set and interval

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Solve each inequality algebraically.x-1x+2≥-2

Short Answer

Step by step solution

Step 1. Change the inequality to equal to zero.

Step 2. Form the intervals

Step 3. Form the table

Step 4. Solution set and interval

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Geometry

Mechanics Maths

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

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