Chapter 4: Q. 44 (page 242)

Solve each inequality algebraically.
$\frac{5}{x - 3} > \frac{3}{x + 1}$

Short Answer

Expert verified

Solution to the inequality $\frac{5}{x - 3} > \frac{3}{x + 1}$ is $(- 7, - 1) \cup (3, \infty)$ .

Step by step solution

Step 1. Given information

We have been given an inequality $\frac{5}{x - 3} > \frac{3}{x + 1}$ .

We have to solve this inequality algebraically.

Step 2. Write the inequality so that a polynomial or rational expression f is on the left side and zero is on the right side

$\frac{5}{x - 3} > \frac{3}{x + 1} \frac{5}{x - 3} - \frac{3}{x + 1} > 0 \frac{5 (x + 1) - 3 (x - 3)}{(x - 3) (x + 1)} > 0 (Take LCD) \frac{5 x + 5 - 3 x + 9}{(x - 3) (x + 1)} > 0 \frac{2 x + 14}{(x - 3) (x + 1)} > 0$

Step 3. Determine the real numbers at which the expression f equals zero and at which the expression f is undefined.

Assume $f (x) = \frac{2 x + 14}{(x - 3) (x + 1)}$ .

$2 x + 14 = 0 2 x = - 14 x = - 7 x - 3 = 0 x = 3 x + 1 = 0 x = - 1$

Step 4. Form the intervals

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

$(- \infty, - 7) \cup (- 7, - 1) \cup (- 1, 3) \cup (3, \infty)$

Step 5. Select a number in each interval and evaluate f at the number

Create the following table:

Interval	$(- \infty, - 7)$	$(- 7, - 1)$	$(- 1, 3)$	$(3, \infty)$
Number chosen	$- 8$	$- 2$	0	4
Value of f	$f (- 8) = - \frac{1}{7}$	$f (- 2) = 1$	$f (0) = - \frac{7}{3}$	$f (4) = \frac{11}{5}$
Conclusion	negative	positive	negative	positive

Step 6. Identify the interval

Since we want to know where f is positive, we conclude that $f (x) > 0$ in the interval $(- 7, - 1) \cup (3, \infty)$ .

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Solve each inequality algebraically.
$\frac{5}{x - 3} > \frac{3}{x + 1}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Write the inequality so that a polynomial or rational expression f is on the left side and zero is on the right side

Step 3. Determine the real numbers at which the expression f equals zero and at which the expression f is undefined.

Step 4. Form the intervals

Step 5. Select a number in each interval and evaluate f at the number

Step 6. Identify the interval

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Solve each inequality algebraically. 5x-3>3x+1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Write the inequality so that a polynomial or rational expression f is on the left side and zero is on the right side

Step 3. Determine the real numbers at which the expression f equals zero and at which the expression f is undefined.

Step 4. Form the intervals

Step 5. Select a number in each interval and evaluate f at the number

Step 6. Identify the interval

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Calculus

Discrete Mathematics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Solve each inequality algebraically.
$\frac{5}{x - 3} > \frac{3}{x + 1}$