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1. Preview Warm Up California Standards Lesson Presentation

3. California Standards 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

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.

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

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

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

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

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

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

14. 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.

15. Additional Example 3A: Identifying the Least Common Multiple Find the LCM of the given expressions. 12x2y, 9xy3 Write the prime factorization of each expression. Align common factors. 9xy3 = 3 3  x  y  y  y 12x2y = 2 2  3  x x  y LCM = 2 2  3  3  x  x  y  y  y = 36x2y3

16. Additional 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) Align common factors. c2 + 8c + 15 = (c + 3) (c + 5) LCM = 3(c + 3)2(c + 5)

17. Check It Out! Example 3a Find the LCM of the given expressions. 5f2h, 15fh2 Write the prime factorization of each expression. Align common factors. 5f2h = 5 ff h 15fh2 = 3 5  f  h  h LCM = 3 5  f fh  h = 15f2h2

18. 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) Align common factors. (x – 6)(x + 5) = (x – 6)(x + 5) LCM = (x– 6)(x + 5)(x + 2)

19. 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.

20. Adding or Subtracting Rational Expressions Step 1 Identify a 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 3 Write each expression using the common denominator. Step 4 Add or subtract the numerators, combining like terms as needed. Step 5 Factor as needed. Step 6 Simplify as needed.

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

23. Multiply the first expression by to get an LCD of w – 5. Step 2 Step 3 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. Write each expression using the LCD.

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

25. 3d3 d Step 1 2d3 = 2 d d  d LCD = 2 3d d  d = 6d3 Step 2 Step 3 Check It Out! Example 4a Add or subtract. Simplify your answer. Identify the LCD. Multiply each expression by an appropriate form of 1. Write each expression using the LCD.

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

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

28. 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.

29. Direction Distance (mi) Rate (mi/h) Distance rate Time (h) = Upstream (against current) 2 x Downstream (with current) 2 3x Additional 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.

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

31. It will take Roland of an hour or 64 minutes to make the round trip. 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.

32. 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.

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

34. total time = Step 4 Step 5 Step 6 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.

35. It will take Katy of an hour or 12.5 minutes to make the round trip. 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.