1. 10-5 Adding and Subtracting Rational Expressions Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 10-5 Adding and Subtracting Rational Expressions Warm Up Add. Simplify your answer. 1. 2. 3.4. Subtract. Simplify your answer. 5. 7. 6. 8.

3. 10-5 Adding and Subtracting Rational Expressions 13.0 Students add, subtract, multiply and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. Also covered:15.0 California Standards

4. 10-5 Adding and Subtracting Rational Expressions 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.

5. 10-5 Adding and Subtracting Rational Expressions

6. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions Check It Out! Example 1a Add. Simplify your answer. = 2 Combine like terms in the numerator. Divide out common factors. Simplify.

10. 10-5 Adding and Subtracting Rational Expressions Check It Out! Example 1b Add. Simplify your answer. Combine like terms in the numerator. Factor. Divide out common factors. Simplify.

11. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions Make sure you add the opposite of each term in the numerator of the second expression when subtracting rational expressions. Caution

13. 10-5 Adding and Subtracting Rational Expressions Check It Out! Example 2a Subtract. Simplify your answer. Subtract numerators. Combine like terms. Factor. Divide out common factors. Simplify.

14. 10-5 Adding and Subtracting Rational Expressions Check It Out! Example 2b Subtract. Simplify your answer. Subtract numerators. Combine like terms. Factor. There are no common factors.

15. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions Check It Out! 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. 10-5 Adding and Subtracting Rational Expressions Check It Out! Example 3b Find the LCM of the given expressions. x 2 – 4x – 12, (x – 6)(x + 5) 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. Check It Out! Example 4a Step 2 Step 3

27. 10-5 Adding and Subtracting Rational Expressions Add or subtract. Simplify your answer. Check It Out! Example 4a Continued Subtract the numerators. Factor and divide out common factors. Step 4 Simplify. Step 5 Step 6

28. 10-5 Adding and Subtracting Rational Expressions Add or subtract. Simplify your answer. Check It Out! 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions 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. 10-5 Adding and Subtracting Rational Expressions Step 2 Use a table to write expressions for time. Check It Out! Example 5 Continued Downstream (with current) Upstream (against current) Rate (mi/h) Distance (mi) Direction Time (h) = distance rate 1 1 4x4x 6x6x

35. 10-5 Adding and Subtracting Rational Expressions Check It Out! 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. 10-5 Adding and Subtracting Rational Expressions 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. Check It Out! Example 5 Continued It will take Katy of an hour or 12.5 minutes to make the round trip.

37. 10-5 Adding and Subtracting Rational Expressions Lesson Quiz: Part I Add or subtract. Simplify your answer. 1. 2. 5. 3. 4.

38. 10-5 Adding and Subtracting Rational Expressions 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