# 7.5 – Adding and Subtracting Rational Expressions

## Key Terms

• Common Denominator – A common multiple of the denominators of two or more fractions.
• Least Common Denominator (LCM) – The smallest common multiple of the denominators of two or more fractions.

## Review

Fractions with the Same Denominator
• To add or subtract fractions or rational expressions that have the same denominator.
1. Add or subtract the numerators.
2. Keep the common denominator.

Fractions with Different Denominators

## Notes

Finding the Common Denominator
•  You can find a common denominator of two rational expressions by multiplying the two denominators together.

• Another way to find the common denominator of two fractions is to find the least common denominator, which is the least common multiple (LCM) of the two denominators.
• How to Use the LCM to Add or Subtract Fractions
1. Find the LCM of the two denominators.
2. Rewrite both fractions using this LCM for both denominators.
3. Add or subtract those fractions.

• LCM – You can also use the least common multiple as a common denominator to add or subtract rational expressions.
1. Factor the denominator of each rational expression.
2. Find the LCM of the denominators.
3. Multiply the numerator and denominator of each rational expression by the factor(s) that make the denominator the LCM.
4. Then you can add or subtract.

• Numbers

• Expressions
• Factor each denominator first!
• Then, ask yourself, “What am I missing from each denominator to make them look the same?”

Denominators with Variables (Exceptions)
• When you have rational expressions as denominators, you need to determine what value(s) x cannot equal.
• Remember, denominators cannot equal 0 (undefined).
• Ex. For $\frac{x^{2}+x-2}{x^{2}(x-2)}$, x ≠ 0 and x ≠ 2.
• To find this, set each factor of the denominator not equal to zero
• $x^{2}\neq 0$ and $x-2\neq 0$ and solve.

## Examples

Rational Expressions with Unlike Denominators
•  Ex 1

Real-World
• Ex 2a. It takes you 45 minutes to wash a car alone. Your friend can wash a car in 30 minutes. How long will it take you and your friend to wash a car together?
• You –> 1 car every 45 min. Friend –> 1 car every 30 min.
• You: $\frac{1}{45}$ + Friend: $\frac{1}{30}$ = Together
• $\frac{1}{45}+\frac{1}{30}=\; Together$
• $(\frac{2}{2})\bullet \frac{1}{45}+(\frac{3}{3})\bullet \frac{1}{30}$
• $\frac{2}{90}+\frac{3}{90}=\frac{5}{90}$
• This reduces by $\frac{5}{5}$, so divide by this to get $\frac{1}{18}$
• Answer: Together, you can wash 1 car every 18 minutes.
• Ex 2b. How many cars per hour is this?  Convert using $\frac{1\; car}{18\; minutes}\bullet \frac{60\; minutes}{1\; hour}=\frac{60}{18}=3.33\; hrs$
• So, that’s about 3 and 1/3 cars per hour.