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Chapter 0: Problem 41

Add or subtract as indicated. $$\frac{3}{x+4}+\frac{6}{x+5}$$

Short Answer

Expert verified
The solution is \(\frac{9x + 39}{(x+4)(x+5)}\)

Step by step solution

01

Find common denominator

The common denominator is the product of the two denominators. So, the common denominator is \( (x+4)(x+5) \).
02

Convert the fractions

Convert the first given fraction, \(\frac{3}{x+4}\), to have the common denominator. Do this by multiplying both the numerator and the denominator by the missing factor from the common denominator, in this case, \(x+5\). Repeat for the second fraction, but multiply by \(x+4\). The result is: \(\frac{3(x+5)}{(x+4)(x+5)}+\frac{6(x+4)}{(x+4)(x+5)}\)
03

Balance the fractions

Now, proceed with the addition: \(\frac{3(x+5) + 6(x+4)}{(x+4)(x+5)}\).
04

Simplify

Simplify the fractions by expanding and collecting like terms. The result is: \(\frac{3x + 15 + 6x + 24}{(x+4)(x+5)}\) which simplifies further to \(\frac{9x + 39}{(x+4)(x+5)}\)

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) toproduce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

Perform the indicated operations. $$ [(3 x+y)+1]^{2} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$

Simplify using properties of exponents. $$\left(x^{\frac{4}{5}}\right)^{5}$$

Explain how to simplify \(\sqrt{10} \cdot \sqrt{5}\)
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