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Chapter 7: Problem 39

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x}{x^{2}-16}+\frac{x}{x-4}$$

Short Answer

Expert verified
The simplified expression is \( \frac{x^2+6x}{(x+4)(x-4)} \).

Step by step solution

01

Factorize the Denominator of the First Fraction

Recognize that the denominator of the first fraction, \(x^{2}-16\), is a difference of two squares. It can be factorized into \((x+4)(x-4)\). Then, rewrite the first fraction as \(\frac{2x}{(x+4)(x-4)}\).
02

Find the Common Denominator

Now take a look at the two fractions. The first fraction has a denominator of \((x+4)(x-4)\) and the second fraction has a denominator of \(x-4\). So, to simplify their addition, they need to have the same denominator. Here, the least common denominator (LCD) is \((x+4)(x-4)\). Multiply the second fraction by \(\frac{x+4}{x+4}\) to achieve the common denominator.
03

Add the Two Fractions

Now that both fractions have the same denominator, they can be added together: \[\frac{2x}{(x+4)(x-4)}+ \frac{x(x+4)}{(x+4)(x-4)}\] Simplify the fractions by adding the numerators and keeping the same denominator. Your result is \(\frac{2x+x(x+4)}{(x-4)(x+4)}\)
04

Simplify the numerator

Expand the expression in the numerator and simplify. This results in \(\frac{2x+x^2+4x}{(x-4)(x+4)}\), which also simplifies to become \(\frac{x^2+6x}{(x-4)(x+4)}\).

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Most popular questions from this chapter

What are similar triangles?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.

Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{3 x+6}{2}-\frac{x}{2}=x+3$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.
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