Problem 104 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identifying the Common Denominator

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

Step 2: 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: \(\frac{3(x + 2)}{n(x + 5)(x + 2)} + \frac{7(x+5)}{(x + 5)(x + 2)}\).

Step 3: 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 \(\frac{3x + 6 + 7x + 35}{(x + 5)(x + 2)}\).