Chapter 8: Q. 267 (page 966)

In the following exercises, simplify.
$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$

Short Answer

Expert verified

The simplified form of the given expression is $\frac{(x + 1) (x - 3)}{2}$ .

Step by step solution

Step 1. Given information.

The given expression is:

$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$

Step 2. Concept.

The following concept is required for simplification:

$\begin{array}{rcl} \frac{A}{B} \div \frac{C}{D} & = & \frac{A}{B} \cdot \frac{D}{C} \end{array}$

Step 3. Rewrite the expression and then simplify.

Simplify the numerator and denominator.

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

Rewrite as the product of first times the reciprocal of the second.

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

Step 4. Conclusion.

The simplified form of the given expression is $\frac{(x + 1) (x - 3)}{2}$ .

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In the following exercises, simplify.
$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept.

Step 3. Rewrite the expression and then simplify.

Step 4. Conclusion.

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In the following exercises, simplify.x-2xx+31x+3+1x-3

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept.

Step 3. Rewrite the expression and then simplify.

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

In the following exercises, simplify.
$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$