1. Homework: Part I Add or subtract. Simplify your answer. 1. 2. 5. 3. 4.

2. Homework: 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.

3. Warm Up Add. Simplify your answer. 1. 2. 3.4. Subtract. Simplify your answer. 5. 7. 6. 8.

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

7. Additional 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. Additional 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. Partner Share! Example 1a Add. Simplify your answer. = 2 Combine like terms in the numerator. Divide out common factors. Simplify.

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

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

12. Make sure you add the opposite of each term in the numerator of the second expression when subtracting rational expressions. Caution

13. Partner Share! Example 2a Subtract. Simplify your answer. Subtract numerators. Combine like terms. Factor. Divide out common factors. Simplify.

14. Partner Share! Example 2b Subtract. Simplify your answer. Subtract numerators. Combine like terms. Factor. There are no common factors.

15. 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 any common multiple of the denominators to find one. You can also use the least common multiple (LCM) of the denominators. To find the LCM of two expressions, write the prime factorization of both expressions. Line up the factors as shown. To find the LCM, multiply one number from each column.

16. Additional Example 3A: Identifying the Least Common Multiple Find the LCM of the given expressions. 12x 2 y, 9xy 3 12x 2 y = 2  2  3  x  x  y  9xy 3 = 3  3  x  y  y  y LCM = 2  2  3  3  x  x  y  y  y Write the prime factorization of each expression. Align common factors. = 36x 2 y 3

17. Additional Example 3B: Identifying the Least Common Multiple Find the LCM of the given expressions. c 2 + 8c + 15, 3c 2 + 18c + 27 c 2 + 8c + 15 = (c + 3) (c + 5) 3c 2 + 18c + 27 = 3(c 2 + 6c +9) = 3(c + 3)(c + 3) LCM = 3(c + 3) 2 (c + 5) Factor each expression. Align common factors.

18. Partner Share! Example 3a Find the LCM of the given expressions. 5f 2 h, 15fh 2 5f 2 h = 5  f  f  h 15fh 2 = 3  5  f  h  h LCM = 3  5  f  f  h  h = 15f 2 h 2 Write the prime factorization of each expression. Align common factors.

19. Partner Share! Example 3b Find the LCM of the given expressions. x 2 – 4x – 12, x 2 – x – 30) x 2 – 4x – 12 = (x – 6) (x + 2) (x – 6)(x + 5) = (x – 6)(x + 5) LCM = (x – 6)(x + 5)(x + 2) Factor each expression. Align common factors.

20. The LCM of the denominators of rational expressions is also called the least common denominator, or LCD, of the rational expressions. You can use the LCD to add or subtract rational expressions.

21. Adding or Subtracting Rational Expressions Step 1 Identify a common denominator. Step 3 Write each expression using the common denominator. Step 2 Multiply each expression by an appropriate form of 1 so that each term has the common denominator as its denominator. Step 4 Add or subtract the numerators, combining like terms as needed. Step 5 Factor as needed. Step 6 Simplify as needed.

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

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

24. Additional 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 Step 3 Multiply the first expression by to get an LCD of w – 5. Write each expression using the LCD.

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

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

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

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

29. Additional 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. Additional Example 5 Continued 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 2 x 3x3x Let x represent the rate of the current, and let 2x represent Roland’s paddling rate.

31. Additional 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 an appropriate form of 1. Write each expression using the LCD, 3x. Add the numerators. Step 4 Step 5 Step 6

32. Additional Example 5 Continued b. 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. Partner Share! 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. Step 2 Use a table to write expressions for time. Partner Share! Example 5 Continued Downstream (with current) Upstream (against current) Rate (mi/h) Distance (mi) Direction Time (h) = distance rate 1 1 4x4x 6x6x

35. Partner Share! Example 5 Continued Step 3 Write and simplify an expression for the total time. total time = time upstream + time downstream Substitute known values. Multiply each fraction by an appropriate form of 1. Write each expression using the LCD, 12x. Add the numerators. total time =Step 4 Step 5 Step 6

36. 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. Partner Share! Example 5 Continued It will take Katy of an hour or 12.5 minutes to make the round trip.

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

38. Lesson Review: 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