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Chapter 7: Problem 104

Explain how to add rational expressions that have different denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

Short Answer

Expert verified
The addition of the given rational expressions is \(\frac{10x + 41}{(x + 5)(x + 2)}\)

Step by step solution

01

Identifying the Common Denominator

The common denominator is found by multiplying the two denominators together. Here, it would be \((x+5)(x+2)\).
02

Modifying Rational Expressions

To get fractions with the same denominator, each fraction is multiplied by an appropriate form of 1 using the missing parts of the common denominator. For the first fraction, we multiply by \(\frac{(x+2)}{(x+2)}\) and the second fraction is multiplied by \(\frac{(x+5)}{(x+5)}\). The expressions become: \(\frac{3(x + 2)}{n(x + 5)(x + 2)} + \frac{7(x+5)}{(x + 5)(x + 2)}\).
03

Adding the Rational Expressions

Now that both fractions have the same denominator, they can be added together as follows \(\frac{3(x + 2)+ 7(x+5)}{(x + 5)(x + 2)}\), this simplifies to \(\frac{3x + 6 + 7x + 35}{(x + 5)(x + 2)}\).

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Most popular questions from this chapter

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{2 y}{y-5}-\left(\frac{2}{y-5}+\frac{y-2}{y-5}\right)$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y+1)(2 y-1)}{(y-2)(y-3)}+\frac{(y+2)(y-1)}{(y-2)(y-3)}-\frac{(y+5)(2 y+1)}{(3-y)(2-y)}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{11}{x+7}-\frac{5}{-x-7}$$

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