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Chapter 7: Problem 40

Add or subtract as indicated. Simplify the result, if possible. $$\frac{4 x}{x^{2}-25}+\frac{x}{x+5}$$

Short Answer

Expert verified
The result after simplifying the expression is \(\frac{x \cdot (x-1)}{(x-5)(x+5)}\).

Step by step solution

01

Identify the denominators

Identify the denominators of the two fractions, which are \(x^{2}-25\) and \(x+5\).
02

Finding the Least Common Denominator (LCD)

In order to add or subtract fractions, we need to have the same denominators. Here, we can factor the first denominator to make it look like the second denominator. \(x^{2}-25\) is a difference of squares and can be factored as \((x-5)(x+5)\).
03

Rewrite the fractions

Now we can write both fractions with the same denominator. Thus, the original equation becomes: \(\frac{4 x}{(x-5)(x+5)}+\frac{(x)(x-5)}{(x-5)(x+5)}\)
04

Simplify

We can combine the fractions and simplify. Continue to simplify by combining like terms in the numerator: \(\frac{4 x+x(x-5)}{(x-5)(x+5)}=\frac{4x+x^{2}-5x}{(x-5)(x+5)}=\frac{x^{2}-x}{(x-5)(x+5)}\)
05

Further simplification

We can simplify the result by factoring the numerator. Factoring \(x^{2}-x\) we get \(x \cdot (x-1)\), so the final answer is \(\frac{x \cdot (x-1)}{(x-5)(x+5)}\)

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used \(\frac{a}{d}=\frac{b}{e}\) to show that corresponding sides of similar triangles are proportional, but I could also use \(\frac{a}{b}=\frac{d}{e}\) or \(\frac{d}{a}=\frac{e}{b}\)

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-8}{x^{2}-16}-\frac{x-8}{16-x^{2}}$$

Anthropologists and forensic scientists classify skulls using $$ \frac{L+60 W}{L}-\frac{L-40 W}{L} $$ where \(L\) is the skull's length and \(W\) is its width. a. Express the classification as a single rational expression. b. If the value of the rational expression in part (a) is less than \(75,\) a skull is classified as long. A medium skull has a value between 75 and \(80,\) and a round skull has a value over \(80 .\) Use your rational expression from part (a) to classify a skull that is 5 inches wide and 6 inches long.

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{y}{a x+b x-a y-b y}-\frac{x}{a x+b x-a y-b y}$$

Factor completely: \(81 x^{4}-1\)
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