1. Splash Screen

2. Five-Minute Check (over Lesson 8–5) CCSS Then/Now New Vocabulary
Example 1: Solve a Rational Equation
Example 2: Solve a Rational Equation
Example 3: Real-World Example: Mixture Problem
Example 4: Real-World Example: Distance Problem
Example 5: Real-World Example: Work Problems
Key Concept: Solving Rational Inequalities
Example 6: Solve a Rational Inequality
Lesson Menu

3. State whether represents a direct, joint,
inverse, or combined variation. Then name the constant of variation.
A. direct;
B. joint;
C. inverse; 2
D. combined; 2
5-Minute Check 1

4. State whether represents a direct, joint,
inverse, or combined variation. Then name the constant of variation.
A. direct;
B. joint;
C. inverse; 2
D. combined; 2
5-Minute Check 1

5. State whether 7.5x = y represents a direct, joint, inverse, or combined variation. Then name the constant of variation.
A. direct; 7.5
B. joint; 7.5
C. inverse;
D. combined; 7.5
5-Minute Check 2

6. State whether 7.5x = y represents a direct, joint, inverse, or combined variation. Then name the constant of variation.
A. direct; 7.5
B. joint; 7.5
C. inverse;
D. combined; 7.5
5-Minute Check 2

7. If y varies inversely as x and y = 8 when x = 12, find y when x = 15.
B. 6.4
C. 8.6
D. 10.2
5-Minute Check 3

8. If y varies inversely as x and y = 8 when x = 12, find y when x = 15.
B. 6.4
C. 8.6
D. 10.2
5-Minute Check 3

9. If y varies jointly as x and z and y = 45 when x = 10 and z = 3, find y when x = 2 and z = 5.
B. 11
C. 13
D. 15
5-Minute Check 4

10. If y varies jointly as x and z and y = 45 when x = 10 and z = 3, find y when x = 2 and z = 5.
B. 11
C. 13
D. 15
5-Minute Check 4

11. A map shows the scale 1. 5 inches equals 65 miles
A map shows the scale 1.5 inches equals 65 miles. How many miles apart are two cities if they are 7.5 inches apart on the map?
A mi
B mi
C. 325 mi
D. 260 mi
5-Minute Check 5

12. A map shows the scale 1. 5 inches equals 65 miles
A map shows the scale 1.5 inches equals 65 miles. How many miles apart are two cities if they are 7.5 inches apart on the map?
A mi
B mi
C. 325 mi
D. 260 mi
5-Minute Check 5

13. The amount of interest earned on a savings account varies jointly with time and the amount deposited. After 5 years, interest on $1000 in the savings account is $225. What is the annual interest rate (constant of variation)?
A. 2%
B. 3.5%
C. 4%
D. 4.5%
5-Minute Check 6

14. The amount of interest earned on a savings account varies jointly with time and the amount deposited. After 5 years, interest on $1000 in the savings account is $225. What is the annual interest rate (constant of variation)?
A. 2%
B. 3.5%
C. 4%
D. 4.5%
5-Minute Check 6

15. Mathematical Practices 6 Attend to precision.
Content Standards
A.CED.1 Create equations and inequalities in one variable and use them to solve problems.
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Mathematical Practices
6 Attend to precision.
CCSS

16. You simplified rational expressions.
Solve rational equations.
Solve rational inequalities.
Then/Now

17. rational equation
weighted average
rational inequality
Vocabulary

18. Solve . Check your solution.
Solve a Rational Equation
Solve Check your solution.
The LCD for the terms is 24(3 – x).
Original equation
Multiply each side by 24(3 – x).
Example 1

19. Distributive Property
Solve a Rational Equation
Distributive Property
Simplify.
Simplify.
Add 6x and –63 to each side.
Example 1

20. The solution is correct. 
Solve a Rational Equation
Check
Original equation
x = –45
Simplify.
Simplify.
The solution is correct. 
Example 1

21. Solve a Rational Equation
Answer:
Example 1

22. Answer: The solution is –45.
Solve a Rational Equation
Answer: The solution is –45.
Example 1

23. Solve
A. –2 B. C.
D. 2
Example 1

24. Solve
A. –2 B. C.
D. 2
Example 1

25. Solve Check your solution.
Solve a Rational Equation
Solve Check your solution.
The LCD is (p + 1)(p – 1).
Original equation
Multiply by the LCD.
Example 2

26. (p – 1)(p2 – p – 5) = (p2 – 7)(p + 1) + p(p + 1)(p – 1)
Solve a Rational Equation
(p – 1)(p2 – p – 5) = (p2 – 7)(p + 1) + p(p + 1)(p – 1)
Divide common factors.
p3 – p2 – 5p – p2 + p + 5 = p3 + p2 – 7p – 7 + p3 – p
Distributive Property
p3 – 2p2 – 4p + 5 = 2p3 + p2 – 8p – 7
Simplify.
0 = p3 + 3p2 – 4p – 12
Subtract p3 – 2p2 – 4p from each side.
Example 2

27. 0 = (p + 3)(p + 2)(p – 2) Factor. 0 = p + 3 or 0 = p + 2 or 0 = p – 2
Solve a Rational Equation
0 = (p + 3)(p + 2)(p – 2)
Factor.
0 = p + 3 or 0 = p + 2 or 0 = p – 2
Zero Product Property
Check Try p = –3.
Original equation ?
p = –3
Example 2

28. Simplify. Simplify. or  Try p = –2. Original equation
Solve a Rational Equation ?
Simplify. ?
Simplify. or 
Try p = –2.
Original equation
Example 2

29. Solve a Rational Equation?
p = –2 ?
Simplify. ?
Simplify. 
Example 2

30. Try p = 2. Original equation p = 2 Simplify. Simplify.  Answer:
Solve a Rational Equation
Try p = 2.
Original equation ?
p = 2 ?
Simplify. ?
Simplify. 
Answer:
Example 2

31. Try p = 2. Original equation p = 2 Simplify. Simplify. 
Solve a Rational Equation
Try p = 2.
Original equation ?
p = 2 ?
Simplify. ?
Simplify. 
Answer: The solutions are –3, –2 and 2.
Example 2

32. A. 4
B. –2
C. 2
D. –4
Example 2

33. A. 4
B. –2
C. 2
D. –4
Example 2

34. Mixture Problem
BRINE Aaron adds an 80% brine (salt and water) solution to 16 ounces of solution that is 10% brine. How much of the solution should be added to create a solution that is 50% brine?
Understand Aaron needs to know how much of a solution needs to be added to an original solution to create a new solution.
Example 3

35. Percentage of brine in solution
Mixture Problem
Plan Each solution has a certain percentage that is salt. The percentage of brine in the final solution must equal the amount of brine divided by the total solution.
Percentage of brine in solution
Example 3

36. Solve Write a proportion.
Mixture Problem
Solve Write a proportion.
Substitute.
Simplify numerator.
LCD is 100(16 + x).
Example 3

37. Divide common factors. Simplify. Distribute. Subtract 50x and 160.
Mixture Problem
Divide common factors.
Simplify.
Distribute.
Subtract 50x and 160.
Divide each side by 30.
Answer:
Example 3

38. Answer: Aaron needs to add ounces of 80% brine solution.
Mixture Problem
Divide common factors.
Simplify.
Distribute.
Subtract 50x and 160.
Divide each side by 30.
Answer: Aaron needs to add ounces of 80% brine solution.
Example 3

39. Check Original equation
Mixture Problem
Check Original equation ? ?
Simplify. 
0.5 = 0.5
Simplify.
Example 3

40. BRINE Janna adds a 65% base solution to 13 ounces of solution that is 20% base. How much of the solution should be added to create a solution that is 40% base?
A. 9.6 ounces
B ounces
C ounces
D ounces
Example 3

41. BRINE Janna adds a 65% base solution to 13 ounces of solution that is 20% base. How much of the solution should be added to create a solution that is 40% base?
A. 9.6 ounces
B ounces
C ounces
D ounces
Example 3

42. Distance Problem
SWIMMING Lilia swims for 5 hours in a stream that has a current of 1 mile per hour. She leaves her dock and swims upstream for 2 miles and then back to her dock. What is her swimming speed in still water?
Understand We are given the speed of the current, the distance she swims upstream, and the total time.
Plan She swam 2 miles upstream against the current and 2 miles back to the dock with the current. The formula that relates distance, time, and rate is d = rt or
Example 4

43. Distance Problem
Let r equal her speed in still water. Then her speed with the current is r + 1, and her speed against the current is r – 1.
Time going with the current
plus
time going against the current
equals
total time. 5
Solve
Original equation
Example 4

44. Multiply each side by r2 – 1.
Distance Problem
Multiply each side by r2 – 1.
Divide Common Factors
(r + 1)2 + (r – 1)2 = 5(r2 – 1)
Simplify.
Distribute.
Simplify.
Subtract 4r from each side.
Example 4

45. Use the Quadratic Formula to solve for r.
Distance Problem
Use the Quadratic Formula to solve for r.
Quadratic Formula
x = r, a = 5, b = – 4, and c = –5
Simplify.
Simplify.
Example 4

46. Distance Problem
r ≈ 1.5 or –0.7
Use a calculator.
Answer:
Example 4

47. Distance Problem
r ≈ 1.5 or –0.7
Use a calculator.
Answer: Since speed must be positive, the answer is about 1.5 miles per hour.
Check
Original equation
r = 1.5 ?
Simplify. ?
Simplify. 
Example 4

48. SWIMMING Lynne swims for 1 hour in a stream that has a current of 2 miles per hour. She leaves her dock and swims upstream for 3 miles and then back to her dock. What is her swimming speed in still water?
A. about 0.6 mph
B. about 2.0 mph
C. about 4.6 mph
D. about 6.6 mph
Example 4

49. SWIMMING Lynne swims for 1 hour in a stream that has a current of 2 miles per hour. She leaves her dock and swims upstream for 3 miles and then back to her dock. What is her swimming speed in still water?
A. about 0.6 mph
B. about 2.0 mph
C. about 4.6 mph
D. about 6.6 mph
Example 4

50. Work Problems
MOWING LAWNS Wuyi and Uima mow lawns together. Wuyi working alone could complete a particular job in 4.5 hours, and Uima could complete it alone in 3.7 hours. How long does it take to complete the job when they work together?
Understand We are given how long it takes Wuyi and Uima working alone to mow a particular lawn. We need to determine how long it would take them together.
Plan Wuyi can mow the lawn in 4.5 hours, so the rate of mowing is of a lawn per hour.
Example 5

51. Work Problems
Uima can mow the lawn in 3.7 hours, so the rate of mowing is of a lawn per hour.
The combined rate is
Example 5

52. Multiply both sides by x.
Work Problems
Solve
Write the equation.
Add
Multiply both sides by x.
x ≈ Multiply 1 by
Answer:
Example 5

53. Multiply both sides by x.
Work Problems
Solve
Write the equation.
Add
Multiply both sides by x.
x ≈ Multiply 1 by
Answer: It would take Wuyi and Uima about 2 hours to mow the lawn together.
Example 5

54. Work Problems
Check
Original equation ?
x ≈ 2
Simplify. 
Example 5

55. A. about 2 hours and 28 minutes B. about 2 hours and 36 minutes
PAINTING Adriana and Monique paint rooms together. Adriana working alone could complete a particular job in 6.4 hours, and Monique could complete it alone in 4.8 hours. How long does it take to complete the job when they work together?
A. about 2 hours and 28 minutes
B. about 2 hours and 36 minutes
C. about 2 hours and 45 minutes
D. about 2 hours and 56 minutes
Example 5

56. A. about 2 hours and 28 minutes B. about 2 hours and 36 minutes
PAINTING Adriana and Monique paint rooms together. Adriana working alone could complete a particular job in 6.4 hours, and Monique could complete it alone in 4.8 hours. How long does it take to complete the job when they work together?
A. about 2 hours and 28 minutes
B. about 2 hours and 36 minutes
C. about 2 hours and 45 minutes
D. about 2 hours and 56 minutes
Example 5

57. Concept

58. Step 2 Solve the related equation.
Solve a Rational Inequality
Solve
Step 1 Values that make the denominator equal to 0 are excluded from the denominator. For this inequality the excluded value is 0.
Step 2 Solve the related equation.
Related equation
Example 6

59. Multiply each side by 9k. Simplify. Add. Divide each side by 6.
Solve a Rational Inequality
Multiply each side by 9k.
Simplify.
Add.
Divide each side by 6.
Example 6

60. Solve a Rational Inequality
Step 3 Draw vertical lines at the excluded value and at the solution to separate the number line into regions.
Now test a sample value in each region to determine if the values in the region satisfy the inequality.
Example 6

61. Test k = –1.  k < 0 is a solution. Solve a Rational Inequality
Example 6

62. 0 < k < is not a solution.
Solve a Rational Inequality
Test k = . 
0 < k < is not a solution.
Example 6

63. Solve a Rational Inequality
Test k = 1. 
Example 6

64. Solve a Rational Inequality
Test k = 1. 
Example 6

65. Solve . A. x < 0 B. x > 0 C. x < 0 or x > 4
D. 0 < x < 4
Example 6

66. Solve . A. x < 0 B. x > 0 C. x < 0 or x > 4
D. 0 < x < 4
Example 6

67. End of the Lesson