1. Splash Screen

2. Five-Minute Check (over Chapter 7) CCSS Then/Now New Vocabulary
Example 1: Simplify a Rational Expression
Example 2: Standardized Test Example: Undefined Values
Example 3: Simplify Using –1
Key Concept: Multiplying Rational Expressions
Example 4: Multiply and Divide Rational Expressions
Example 5: Polynomials in the Numerator and Denominator
Example 6: Simplify Complex Fractions
Lesson Menu

3. Evaluate log12 7. A B C D
5-Minute Check 1

4. Evaluate log12 7. A B C D
5-Minute Check 1

5. A. B. C. D.
5-Minute Check 2

6. A. B. C. D.
5-Minute Check 2

7. Solve log3 (x2 – 12) = log3 4x.
A. 6
B. 7
C. 8
D. 9
5-Minute Check 3

8. Solve log3 (x2 – 12) = log3 4x.
A. 6
B. 7
C. 8
D. 9
5-Minute Check 3

9. Solve 5ex – 3 = 0.
A. –0.5108
B. –0.2197 C D
5-Minute Check 4

10. Solve 5ex – 3 = 0.
A. –0.5108
B. –0.2197 C D
5-Minute Check 4

11. Suppose $200 was deposited in a bank account and it is now worth $1100
Suppose $200 was deposited in a bank account and it is now worth $1100. If the annual interest rate was 5% compounded continuously, how long ago was the account started? Use the formula A = Pert.
A. about 42 years ago
B. about 34 years ago
C. exactly 29 years ago
D. about 24 years ago
5-Minute Check 5

12. Suppose $200 was deposited in a bank account and it is now worth $1100
Suppose $200 was deposited in a bank account and it is now worth $1100. If the annual interest rate was 5% compounded continuously, how long ago was the account started? Use the formula A = Pert.
A. about 42 years ago
B. about 34 years ago
C. exactly 29 years ago
D. about 24 years ago
5-Minute Check 5

13. Suppose the population of New York State grows at a rate of 0
Suppose the population of New York State grows at a rate of 0.3% compounded continuously. In 2006, the population was 19.3 million. Write an equation that represents the population and predict the population in after t years 2020.
A. y = 19.3e(0.003)t; about 20.1 million
B. y = 19.3e(0.03)t; about 29.4 million
C. y = 19.3e(1.003)t; about 52.6 million
D. y = 19.3e(1.3)t; about 70.8 million
5-Minute Check 6

14. Suppose the population of New York State grows at a rate of 0
Suppose the population of New York State grows at a rate of 0.3% compounded continuously. In 2006, the population was 19.3 million. Write an equation that represents the population and predict the population in after t years 2020.
A. y = 19.3e(0.003)t; about 20.1 million
B. y = 19.3e(0.03)t; about 29.4 million
C. y = 19.3e(1.003)t; about 52.6 million
D. y = 19.3e(1.3)t; about 70.8 million
5-Minute Check 6

15. Mathematical Practices
Content Standards
A.APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Mathematical Practices
8 Look for and express regularity in repeated reasoning.
CCSS

16. You factored polynomials.
Simplify rational expressions.
Simplify complex fractions.
Then/Now

17. rational expression
complex fraction
Vocabulary

18. Look for common factors.
Simplify a Rational Expression
A. Simplify
Look for common factors.
Eliminate common factors. ●
Simplify.
Answer:
Example 1A

19. Look for common factors.
Simplify a Rational Expression
A. Simplify
Look for common factors.
Eliminate common factors. ●
Simplify.
Answer:
Example 1A

20. B. Under what conditions is the expression undefined?
Simplify a Rational Expression
B. Under what conditions is the expression undefined?
Just as with a fraction, a rational expression is undefined if the denominator equals zero.
The original factored denominator is (y + 7)(y – 3)(y + 3).
Answer:
Example 1B

21. B. Under what conditions is the expression undefined?
Simplify a Rational Expression
B. Under what conditions is the expression undefined?
Just as with a fraction, a rational expression is undefined if the denominator equals zero.
The original factored denominator is (y + 7)(y – 3)(y + 3).
Answer: The values that would make the denominator equal to 0 are –7, 3, and –3. So the expression is undefined at y = –7, y = 3, and y = –3.
Example 1B

22. A. Simplify A. B. C. D.
Example 1A

23. A. Simplify A. B. C. D.
Example 1A

24. B. Under what conditions is the expression undefined?
A. x = 4 or x = –4
B. x = –5 or x = 4
C. x = –5, x = 4, or x = –4
D. x = –5
Example 1B

25. B. Under what conditions is the expression undefined?
A. x = 4 or x = –4
B. x = –5 or x = 4
C. x = –5, x = 4, or x = –4
D. x = –5
Example 1B

26. For what value(s) of p is undefined? A 5 B –3, 5 C 3, –5 D 5, 1, –3
Undefined Values
For what value(s) of p is undefined?
A 5 B –3, 5
C 3, –5 D 5, 1, –3
Read the Test Item
You want to determine which values of p make the denominator equal to 0.
Example 2

27. p2 – 2p –15 = (p – 5)(p + 3) Factor the denominator.
Undefined Values
Solve the Test Item
Look at the possible answers. Notice that the p term and the constant term are both negative, so there will be one positive solution and one negative solution. Therefore, you can eliminate choices A and D. Factor the denominator.
p2 – 2p –15 = (p – 5)(p + 3) Factor the denominator.
p – 5 = 0 or p + 3 = 0 Zero Product Property
p = 5 p = –3 Solve each equation.
Answer:
Example 2

28. p2 – 2p –15 = (p – 5)(p + 3) Factor the denominator.
Undefined Values
Solve the Test Item
Look at the possible answers. Notice that the p term and the constant term are both negative, so there will be one positive solution and one negative solution. Therefore, you can eliminate choices A and D. Factor the denominator.
p2 – 2p –15 = (p – 5)(p + 3) Factor the denominator.
p – 5 = 0 or p + 3 = 0 Zero Product Property
p = 5 p = –3 Solve each equation.
Answer: B
Example 2

29. For what value(s) of p is undefined?
B. –5
C. 5
D. –5, –3
Example 2

30. For what value(s) of p is undefined?
B. –5
C. 5
D. –5, –3
Example 2

31. Factor the numerator and the denominator.
Simplify Using –1
Simplify
Factor the numerator and the denominator.
b – 2 = –(–b + 2) or –1(2 – b)
Simplify.
Answer:
Example 3

32. Factor the numerator and the denominator.
Simplify Using –1
Simplify
Factor the numerator and the denominator.
b – 2 = –(–b + 2) or –1(2 – b)
Simplify.
Answer: –a
Example 3

33. Simplify
A. y – x
B. y
C. x
D. –x
Example 3

34. Simplify
A. y – x
B. y
C. x
D. –x
Example 3

35. Concept

36. A. Simplify . Simplify. Simplify. Answer:
Multiply and Divide Rational Expressions
A. Simplify
Simplify.
Simplify.
Answer:
Example 4A

37. A. Simplify . Simplify. Simplify. Answer:
Multiply and Divide Rational Expressions
A. Simplify
Simplify.
Simplify.
Answer:
Example 4A

38. Multiply by the reciprocal of the divisor.
Multiply and Divide Rational Expressions
B. Simplify
Multiply by the reciprocal of the divisor.
Simplify.
Example 4B

39. Multiply and Divide Rational Expressions
Simplify.
Answer:
Example 4B

40. Multiply and Divide Rational Expressions
Simplify.
Answer:
Example 4B

41. A. Simplify A. B. C. D.
Example 4A

42. A. Simplify A. B. C. D.
Example 4A

43. B. Simplify
A. AnsA
B. AnsB
C. AnsC
D. AnsD
Example 4B

44. B. Simplify
A. AnsA
B. AnsB
C. AnsC
D. AnsD
Example 4B

45. A. Simplify . Factor. 1 + k = k + 1, 1 – k = –1(k – 1) = –1 Simplify.
Polynomials in the Numerator and Denominator
A. Simplify
Factor.
1 + k = k + 1, 1 – k = –1(k – 1)
= –1 Simplify.
Answer:
Example 5A

46. A. Simplify . Factor. 1 + k = k + 1, 1 – k = –1(k – 1) = –1 Simplify.
Polynomials in the Numerator and Denominator
A. Simplify
Factor.
1 + k = k + 1, 1 – k = –1(k – 1)
= –1 Simplify.
Answer: –1
Example 5A

47. Multiply by the reciprocal of the divisor.
Polynomials in the Numerator and Denominator
B. Simplify
Multiply by the reciprocal of the divisor.
Factor.
Example 5B

48. Simplify. Answer: Polynomials in the Numerator and Denominator
Example 5B

49. Simplify. Answer: Polynomials in the Numerator and Denominator
Example 5B

50. A. Simplify A. B.
C. 1
D. –1
Example 5A

51. A. Simplify A. B.
C. 1
D. –1
Example 5A

52. A. B. C. D.
Example 5B

53. A. B. C. D.
Example 5B

54. Express as a division expression.
Simplify Complex Fractions
Simplify
Express as a division expression.
Multiply by the reciprocal of the divisor.
Example 6

55. Simplify Complex Fractions
Factor. –1
Simplify.
Answer:
Example 6

56. Simplify Complex Fractions
Factor. –1
Simplify.
Answer:
Example 6

57. Simplify
A. e B.
C. e D.
Example 6

58. Simplify
A. e B.
C. e D.
Example 6

59. End of the Lesson