Splash Screen

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

Evaluate log12 7. A. 2.4849 B. 1.9459 C. 0.7831 D. 0.5389 5-Minute Check 1

Evaluate log12 7. A. 2.4849 B. 1.9459 C. 0.7831 D. 0.5389 5-Minute Check 1

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

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

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

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

Solve 5ex – 3 = 0. A. –0.5108 B. –0.2197 C. 0.2197 D. 0.4979 5-Minute Check 4

Solve 5ex – 3 = 0. A. –0.5108 B. –0.2197 C. 0.2197 D. 0.4979 5-Minute Check 4

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

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

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

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

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

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

rational expression complex fraction Vocabulary

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Concept

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A. B. C. D. Example 5B

A. B. C. D. Example 5B

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

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

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

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

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

End of the Lesson