Slide 1 :
Slide 2 : Lesson 11-1 Inverse Variation
Lesson 11-2 Rational Expressions
Lesson 11-3 Multiplying Rational Expressions
Lesson 11-4 Dividing Rational Expressions
Lesson 11-5 Dividing Polynomials
Lesson 11-6 Rational Expressions with Like Denominators
Lesson 11-7 Rational Expressions with Unlike Denominators
Lesson 11-8 Mixed Expressions and Complex Fractions
Lesson 11-9 Rational Equations and Functions
Slide 3 : Five-Minute Check (over Lesson 11-7)
Main Ideas and Vocabulary
California Standards
Example 1: Mixed Expression to Rational Expression
Key Concept: Simplifying a Complex Fraction
Example 2: Complex Fraction Involving Numbers
Example 3: Complex Fraction Involving Monomials
Example 4: Complex Fraction Involving Polynomials
Slide 4 : mixed expression complex fraction Simplify mixed expressions. Simplify complex fractions.
Slide 5 : Standard 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. (Key)
Standard 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. (Key)
Slide 6 : Mixed Expression to Rational Expression The LCD is x – 2. Add the numerators. Distributive Property Simplify. Answer:
Slide 7 : A
B
C
D
Slide 8 :
Slide 9 : To find the total number of cookies, divide the amount of cookie dough by the amount of dough needed for each cookie. Complex Fraction Involving Numbers
Slide 10 : Complex Fraction Involving Numbers Convert pounds to ounces and divide by common units. Simplify. Express each term as an improper fraction.
Slide 11 : Answer: Katelyn can make 21 cookies. Complex Fraction Involving Numbers Simplify.
Slide 12 : A
B
C
D A. 27 cookies
B. 30 cookies
C. 25 cookies
D. 20 cookies
Slide 13 : Complex Fraction Involving Monomials Rewrite as a division sentence. Rewrite as multiplication by the reciprocal.
Slide 14 : Complex Fraction Involving Monomials Divide by common factors a, b, and c2. Simplify. Answer:
Slide 15 : A
B
C
D
Slide 16 : The numerator contains a mixed expression. Rewrite it as a rational expression first. Complex Fraction Involving Polynomials The LCD of the fractions in the numerator is b + 3. Simplify the numerator.
Slide 17 : Complex Fraction Involving Polynomials Factor. Multiply by the reciprocal of b – 4. Simplify. Rewrite as a division sentence. Answer:
Slide 18 : A
B
C
D
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Slide 20 : Five-Minute Check (over Lesson 11-8)
Main Ideas and Vocabulary
California Standards
Example 1: Use Cross Products
Example 2: Use the LCD
Example 3: Rational Functions
Example 4: Work Problem
Example 5: Rate Problem
Example 6: Extraneous Solutions
Slide 21 : rational equations work problems
rate problems
extraneous solutions Solve rational equations. Eliminate extraneous solutions.
Slide 22 : Standard 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. (Key)
Standard 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. (Key, CAHSEE)
Slide 23 : Use Cross Products x = 9 Original equation Cross multiply. Distributive Property Add –2x and 48 to each side. 8x – 48 = 2x + 6 6x = 54 Divide each side by 6. 8(x – 6) = 2(x + 3) Answer: 9
Slide 24 : A
B
C
D A. 3
B. 0
C. –3
D. 6
Slide 25 : Use the LCD Original equation The LCD is x(x + 1).
Slide 26 : Use the LCD Distributive Property Simplify. Add. 4x – 1 = 2 5x – (x + 1) = 2 Add 1 to each side. 4x = 3
Slide 27 : Use the LCD Answer: Divide each side by 4. Simplify.
Slide 28 : Use the LCD Original equation The LCD is a2 – 1. Distributive Property
Slide 29 : Use the LCD Simplify. Set equal to 0. Factor. a2 – 2a – 3 = 0 (a – 3)(a + 1) = 0 a3 – a + (a2 – 5) = a3 + a – 2 a – 3 = 0 or a + 1 = 0 a = 3 a = –1 Zero Product Property Simplify. Answer: If a = –1, then a + 1 = 0. Since division by 0 is undefined, there is only solution: a = 3.
Slide 30 : A
B
C
D A. 1
B. –2
C. 4
D. 8
Slide 31 : A
B
C
D A. 4, –1
B. 4
C. –1
D. 2, 4, –1
Slide 32 : Rational Functions Answer: When x = 2 and –8, the numerator becomes zero, so f(x) = 0. Therefore, the roots of the function are 2 and –8.
Slide 33 : A
B
C
D A. 1, 2
B. –2, 1
C. –1, 2
D. –1, –1
Slide 34 : Work Problem
Slide 35 : Work Problem Explore
Slide 36 : Work Problem Solve Lee’s her father’s total work plus work equals work.
Slide 37 : Work Problem Multiply. The LCD is 10t. Distributive Property Simplify.
Slide 38 : Work Problem Check This seems reasonable because the combined efforts of the two took longer than half of her father’s usual time. Add –6t to each side. Divide each side by 4.
Slide 39 : A
B
C
D
Slide 40 : TRANSPORTATION The schedule for the Washington, D.C., Metrorail is shown to the right. Suppose two Red Line trains leave their stations at opposite ends of the line at exactly 2:00 P.M. One train travels between the two stations in 48 minutes and the other train takes 54 minutes. At what time do the two trains pass each other? Rate Problem
Slide 41 : Determine the rates of both trains. The total distance is 19.4 miles. Next, since both trains left at the same time, the time both have traveled when they pass will be the same. And since they started at opposite ends of the route, the sum of their distances is equal to the total route, 19.4 miles. Rate Problem
Slide 42 : Rate Problem The sum of the distances is 19.4 The LCD is 432.
Slide 43 : Answer: The trains passed each other about 25 minutes after they left their stations, at 2:25 P.M. Rate Problem Simplify. Add. Divide each side by 329.8 Distributive Property Animation: Rate Problem
Slide 44 : A
B
C
D A. 3:27 P.M.
B. 3:30 P.M.
C. 3:50 P.M.
D. 4:00 P.M. TRANSPORTATION Two cyclists are riding on a 5-mile circular bike trail. They both leave the bike trail entrance at 3:00 P.M. traveling in opposite directions. It usually takes the first cyclist one hour to complete the trail and it takes the second cyclist 50 minutes. At what time will they pass each other?
Slide 45 : Extraneous Solutions Original equation The LCD is x – 1. Distributive Property Simplify. Answer: no solutions
Slide 46 : A
B
C
D A. 2, –3
B. –3
C. infinitely many solutions
D. no solution Interactive Lab: Exploring Rational Expressions and Equations
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Slide 48 : 11-5 Use Algebra Tiles to Find (x2 + 3x + 2) ÷ (x + 1)
11-9 Rate Problem
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