1. Adding and Subtracting Rational Expressions 12-5
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1

2. Warm Up
Add. Simplify your answer.
Subtract. Simplify your answer. 5. 6. 7. 8.

3. Objectives
Add and subtract rational expressions with like denominators.
Add and subtract rational expressions with unlike denominators.

4. The rules for adding rational expressions are the same as the rules for adding fractions. If the denominators are the same, you add the numerators and keep the common denominator.

6. Example 1A: Adding Rational Expressions with Like Denominators
Add. Simplify your answer.
Combine like terms in the numerator. Divide out common factors.
Simplify.

7. Example 1B: Adding Rational Expressions with Like Denominators
Add. Simplify your answer.
Combine like terms in the numerator.
Factor. Divide out common factors.
Simplify.

8. Example 1C: Adding Rational Expressions with Like Denominators
Add. Simplify your answer.
Combine like terms in the numerator.
Factor. Divide out common factors.
Simplify.

9. Check It Out! Example 1a
Add. Simplify your answer.
Combine like terms in the numerator. Divide out common factors.
= 2
Simplify.

10. Check It Out! Example 1b
Add. Simplify your answer.
Combine like terms in the numerator.
Factor. Divide out common factors.
Simplify.

11. To subtract rational expressions with like denominators, remember to add the opposite of each term in the second numerator.

12. Example 2: Subtracting Rational Expressions with Like Denominators
Subtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. Divide out common factors.
Simplify.

13. Make sure you add the opposite of all the terms in the numerator of the second expression when subtracting rational expressions.
Caution

14. Check It Out! Example 2a
Subtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. Divide out common factors.
Simplify.

15. Check It Out! Example 2b
Subtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. There are no common factors.

16. As with fractions, rational expressions must have a common denominator before they can be added or subtracted. If they do not have a common denominator, you can use the least common multiple, or LCM, of the denominators to find one.
To find the LCM, write the prime factorization of both expressions. Use each factor the greatest number of times it appears in either expression.

17. Example 3A: Identifying the Least Common Multiple
Find the LCM of the given expressions.
12x2y, 9xy3
Write the prime factorization of each expression. Use every factor of both expressions the greatest number of times it appears in either expression.
9xy3 =  3  x  y  y  y
12x2y = 2  2  3  x  x  y
LCM = 2  2  3  3  x  x  y  y  y
= 36x2y3

18. Example 3B: Identifying the Least Common Multiple
Find the LCM of the given expressions.
c2 + 8c + 15, 3c2 + 18c + 27
Factor each expression.
3c2 + 18c + 27 = 3(c2 +6c +9)
= 3(c + 3)(c + 3)
Use every factor of both expressions the greatest number of times it appears in either expression.
c2 + 8c + 15 = (c + 3) (c + 5)
LCM = 3(c + 3)2(c + 5)

19. Check It Out! Example 3a
Find the LCM of the given expressions.
5f2h, 15fh2
Write the prime factorization of each expression. Use every factor of both expressions the greatest number of times it appears in either expression.
5f2h =  f  f  h
15fh2 = 3  5  f  h  h
LCM = 3  5  f  f  h  h
= 15f2h2

20. Check It Out! Example 3b
Find the LCM of the given expressions.
x2 – 4x – 12, (x – 6)(x +5)
Factor each expression.
x2 – 4x – 12 = (x – 6) (x + 2)
Use every factor of both expressions the greatest number of times it appears in either expression.
(x – 6)(x +5) = (x – 6)(x + 5)
LCM = (x – 6)(x + 5)(x + 2)

21. The LCM of the denominators of fractions or rational expressions is also called the least common denominator, or LCD. You use the same method to add or subtract rational expressions.

22. Example 4A: Adding and Subtracting with Unlike Denominators
Add or subtract. Simplify your answer.
Identify the LCD.
5n3 =  n  n  n
Step 1
2n2 =  n  n
LCD = 2  5  n  n  n = 10n3
Multiply each expression by an appropriate form of 1.
Step 2
Step 3
Write each expression using the LCD.

23. Example 4A Continued
Add or subtract. Simplify your answer.
Step 4
Add the numerators.
Step 5
Factor and divide out common factors.
Step 6
Simplify.

24. Example 4B: Adding and Subtracting with Unlike Denominators.
Add or subtract. Simplify your answer.
Step 1 The denominators are opposite binomials. The LCD can be either w – 5 or 5 – w.
Identify the LCD.
Step 2
Multiply the first expression by to get an LCD of w – 5.
Step 3
Write each expression using the LCD.

25. Example 4B Continued
Add or Subtract. Simplify your answer.
Step 4
Subtract the numerators.
Step 5, 6
No factoring needed, so just simplify.

26. Check It Out! Example 4a
Add or subtract. Simplify your answer.
3d  d
2d3 = 2  d  d  d
LCD = 2  3 d  d  d = 6d3
Step 1
Identify the LCD.
Multiply each expression by an appropriate form of 1.
Step 2
Step 3
Write each expression using the LCD.

27. Check It Out! Example 4a Continued
Add or subtract. Simplify your answer.
Step 4
Subtract the numerators.
Step 5
Factor and divide out common factors.
Step 6
Simplify.

28. Check It Out! Example 4b
Add or subtract. Simplify your answer.
Step 1
Factor the first term. The denominator of second term is a factor of the first.
Step 2
Add the two fractions.
Step 3
Divide out common factors.
Step 4
Simplify.

29. Example 5: Recreation Application
Roland needs to take supplies by canoe to some friends camping 2 miles upriver and then return to his own campsite. Roland’s average paddling rate is about twice the speed of the river’s current.
a. Write and simplify an expression for how long it will take Roland to canoe round trip.
Step 1 Write expressions for the distances and rates in the problem. The distance in both directions is 2 miles.

30. Step 2 Use a table to write expressions for time.
Example 5 Continued
Let x represent the rate of the current, and let 2x represent Roland’s paddling rate.
Roland’s rate against the current is 2x – x, or x. Roland’s rate with the current is 2x + x, or 3x.
Step 2 Use a table to write expressions for time.
Downstream
(with current)
Upstream
(against current)
Rate
(mi/h)
Distance (mi)
Direction
Time (h) =
Distance
rate 2 x 3x

31. Example 5 Continued
Step 3 Write and simplify an expression for the total time.
total time = time upstream + time downstream
total time =
Substitute known values.
Multiply the first fraction by a appropriate form of 1.
Step 4
Step 5
Write each expression using the LCD 3x.
Step 6
Add the numerators.

32. Example 5 Continued
b. If the speed of the river’s current is 2.5 miles per hour, about how long will it take Roland to make the round trip?
Substitute 2.5 for x. Simplify.
It will take Roland of an hour or 64 minutes to make the round trip.

33. Check It Out! Example 5
What if?...Katy’s average paddling rate increases to 5 times the speed of the current. Now how long will it take Katy to kayak the round trip?
Step 1 Let x represent the rate of the current, and let 5x represent Katy’s paddling rate.
Katy’s rate against the current is 5x – x, or 4x. Katy’s rate with the current is 5x + x, or 6x.

34. Check It Out! Example 5 Continued
Step 2 Use a table to write expressions for time.
Downstream
(with current)
Upstream
(against current)
Rate
(mi/h)
Distance (mi)
Direction
Time (h) =
distance
rate 1 4x 6x

35. Check It Out! Example 5 Continued
Step 3 Write and simplify an expression for the total time.
total time = time upstream + time downstream
total time =
Substitute known values.
Multiply each fraction by a appropriate form of 1.
Step 4
Step 5
Write each expression using the LCD 24x.
Step 6
Add the numerators.

36. Check It Out! Example 5 Continued
b. If the speed of the river’s current is 2 miles per hour, about how long will it take Katy to make the round trip?
Substitute 2 for x. Simplify.
It will take Katy of an hour or 12.5 minutes to make the round trip.

37. Lesson Quiz: Part I
Add or subtract. Simplify your answer. 1. 2. 3. 4. 5.

38. Lesson Quiz: Part II
6. Vong drove 98 miles on interstate highways and 80 miles on state roads. He drove 25% faster on the interstate highways than on the state roads. Let r represent his rate on the state roads in miles per hour.
a. Write and simplify an expression that represents the number of hours Vong drove in terms of r.
b. Find Vong’s driving time if he averaged 55 miles per hour on the state roads.
about 2 h 53 min