Mini-Lecture 6.1 Rational Functions and Multiplying and Dividing Rational Expressions
|
|
- Meryl Carpenter
- 5 years ago
- Views:
Transcription
1 Learning Objectives: Mini-Lecture 6. Rational Functions and Multiplying and Dividing Rational Epressions. Find the domain of a rational function.. Simplify rational epressions.. Multiply rational epressions. 4. Divide rational epressions. 5. Use rational functions in applications. 6. Key vocabulary: undefined, domain, simplify, reciprocal. Eamples:. Find the domain of each rational epression a) f( ) = b) g ( ) = c) h ( ) = 7 + d) 9 p ( ) = e) f( ) = +. Simplify each rational epression. a) 6 b) c) d) 8 8 e) y y + 5y y + f) y 64 4y 6. Multiply and simplify. 8 a) 5 5 b) 4y 49 8 y c) Divide and simplify. 4 a) b) c) Use rational functions in applications. A company s cost per book for printing particular books is given by the rational functions C( ) =. Find the cost per book for printing a) 00 books b) 000 books Teaching Notes: Many students need a review of simplifying, multiplying and dividing numerical fractions before attempting algebraic ones. Many students have trouble with problems where the factors in the numerator and denominator have opposite signs. Refer to the end-of-section eercises for applied problems. Refer students to the Simplifying/ Multiplying/ Dividing Rational Epressions chart in the tet. Answers: a) { is a real number}, b) { is a real number and 0}, c) { is a real number and 7}, d) { is a real number and 5 }, e) { is a real number and y +, }; a) -, b) -9, c) -5, d) -, e) 7 y + 7, f) y + 4y+ 6 4 ; a) 4 9, b), c) 5 + ( 7) 9 ; 4a) 5, b) + 4, c) ; 5a) $7.47, b) $.47 + M - Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
2 Mini-Lecture 6. Adding and Subtracting Rational Epressions Learning Objectives:. Add or subtract rational epressions with common denominators.. Identify the Least Common Denominator (LCD) of two or more rational epressions.. Add or subtract rational epressions with unlike denominators. 4. Key vocabulary: least common denominator. Eamples:. Add or subtract as indicated. a) 8 + b) c). Find the LCD of the rational epressions in each list. a) c) ' 7 4 9, + 6 b), 7y y 6 5 d), ' y + y+ y 8. Add or subtract as indicated. If possible, simplify your answer. a) b) 6y 5y + c) 6 + r r 8 d) e) f) + +. Teaching Notes: 0 6 g) Most students need a review of adding, subtracting, and finding LCDs of numerical fractions before attempting algebraic ones. Many students find this section difficult. Some students need to see more eamples for objective. Etra time spent here is well worth it and pays off with greater success in objective. Refer students to the Adding or Subtracting Rational Epressions with Common/Different Denominators and Finding the Least Common Denominator charts in the tet. Answers: a), b) -, c) ; a) 77, b) 7y, c) (-)(+), d) 8(+y) (-y); a) + 4 d) 4, e) ( )( + 6)( + 8), f) 7 6, g) ( + 5) 9 0y 7+, b), c) 4r, rr ( ) M - Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
3 Mini-Lecture 6. Simplifying Comple Fractions Learning Objectives:. Simplify comple fractions by simplifying the numerator and denominator and then dividing.. Simplify comple fractions by multiplying by a common denominator.. Simplify epressions with negative eponents. 4. Key vocabulary: comple rational epression. Eamples:. Simplify each comple fraction by simplifying the numerator and denominator and then dividing. a) b) c) a 9 9 a d) 6 5y y 4 5 y e) f) g) h) Simplify selected problems from a) through h) by multiplying the least common denominator.. Simplify.. a) + y b) + 5y 7 y c) 6 + (6 y) Teaching Notes: Stronger students tend to prefer using the multiply by LCD method. Many students need to be reminded of how to deal with negative eponents before attempting objective. Refer students to the Simplifying a Comple Fraction: Method / Method charts in the tet. Answers: a) 5 7, b) + a + 9, c), d) 4+5y, e) 4 a 4 8 b) y + 5 y 6y+, c) 7y 6y, f) 5 0, g) 4 0, h) 6; a-h) same as a-h); a) + y+, y M-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
4 Learning Objectives: Mini-Lecture 6.4 Dividing Polynomials: Long Division and Synthetic Division. Divide a polynomial by a monomial.. Divide by a polynomial.. Use synthetic division to divide a polynomial by a binomial. 4. Use the remainder theorem to evaluate polynomials. Eamples:. Divide. a) by 4 b) y + 9 y y y c) y + y 6 y 4 4 y. Divide. ) ( a) ( 5 5 b) ) ( ) ( ) ( c) ( d) ) + ) ( 0+ + ) ( 6 ). Use synthetic division to divide. a) b) c) d) Use the remainder theorem to P(c). a) P() = + 8; b) P() = ; Teaching Notes: Remind students to check their answers by multiplying. Encourage students to write the intermediate step a + b = a + b when dividing by a monomial. c c c Most students understand dividing by a binomial better if a numerical long division problem is shown in parallel. Most students understand the synthetic division process after a couple of eamples. Most students prefer synthetic division over long division. Answers: a) -, b) +y-y 4y, c) 8+ ; a) +7, b) 5-7, c) - -5+, d) --; a) -9, b) +5, c) - ++6, d) a), b) 08 M-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
5 Mini-Lecture 6.5 Solving Equations Containing Rational Epressions Learning Objectives:. Solve equations containing rational epressions... Key vocabulary: equation versus epression, etraneous solutions. Eamples:. Solve each equation and check the solution. a) y y= 5 b) 5 y + 6 = + y 5 4 c) 4 = 5 d) 5 = e) + = f) 4+ 6 = 5 6 g) 4 = = h) 5 i) = 5 5. Teaching Notes: j) + = Remind students to always determine the values not allowed for before solving a rational epression. Many students are confused by the concept of an etraneous solution. Show them a simple eample such as : = = = = 0 = 0,; =0is etraneous. Some students prefer to make equivalent fractions with a common denominator, and then set the numerators equal to each other. Refer students to the To Solve an Equation Containing Rational Epressions chart in the tet. Answers: a) {75}, b) 4 ; c) {}, d) {7}, e) {-5,4}, f) {}, g) 5, h) {}; i), j) M-6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
6 Mini-Lecture 6.6 Rational Equations and Problem Solving Learning Objectives:. Solve an equation containing rational epressions for a specified variable.. Solve problems by writing equations containing rational epressions.. Key vocabulary: ratio, rate, proportion. Eamples:. Solve each equation for the specific variable. a) PV T pv = for V b) t + = for c c) a b c A P = for r + rt d) GMm A = h ( B+ b ) for B e) F = for M f) r a anr S = r for a. Solve. a) Number Two times the reciprocal of a number equals 4 times the reciprocal of 5. Find the number. b) Proportion The ratio of the weight of an object on Earth to an object on Planet X is 4 to 9. If a person weighs 0 pounds on Earth, find his weight on planet X. Round to the nearest whole number. c) Work One pump can drain a pool in 9 minutes. When a second pump is also used, the pool only takes 6 minutes to drain. How long would it take the second pump to drain the pool if it were the only pump in use? d) Rate Ale can run 5 miles per hour on level ground on a still day. One windy day he runs miles with the wind, and in the same amount of time runs 4 miles against the wind. What is the rate of the wind? Teaching Notes: Many students find this section difficult. Most students need to set up a chart to solve work and rate problems. Refer them to the tetbook eamples for samples. Encourage students to check whether their solutions seem reasonable. Refer to students to the Solving an Equation for a Specified Variable chart in the tet. pvt ab Answers: a) V =, b) c tp = b + a, c) A P r =, d) Pt b) 58 pounds, c) 8 minutes, d) mph A bh B =, e) h Fr M =, f) a = S( r) + anr ; a) Gm 5, M-7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
7 Mini-Lecture 6.7 Variation and Problem Solving Learning Objectives:. Solve problems involving direct variation.. Solve problems involving inverse variation.. Solve problems involving joint variation. 4. Solve problems involving combined variation. 5. Key vocabulary: constant of variation or constant of proportionality. Eamples:. Find the constant of variation and the direct variation equation for each situation. Then solve as indicated. a) y = 4 when =. Find y when = 9. b) The amount of gas that a helicopter uses is directly proportional to the number of hours spent flying. The helicopter flies for hours and uses 8 gallons of fuel. Find the number of gallons of fuel that the helicopter uses to fly for 5 hours.. Find the constant of variation and the inverse variation equation for each situation. Then solve as indicated. a) y = 4 when =. Find y when = 6. b) The amount of time it takes a swimmer to swim a race is inversely proportional to the swimmer s speed. A swimmer finishes a race in 50 seconds with a speed of feet per second. Find the speed if it takes 5 seconds to finish the race.. Find the constant of variation and the joint or the combined variation equation for each situation. Then solve as indicated. a) r varies jointly as the square of s and the square of t. r = when s = and t =. b) is directly proportional to y and inversely proportional to the cube of z. = when y = and z =. c) The volume V of a given mass of gas varies directly as the temperature T and inversely as the pressure P. A measuring device is calibrated to give V = 00 in when T = 50 and P= 0 lb/ in. What is the volume on this device when the temperature is 70 and the pressure is 0 lb / in? Teaching Notes: Most students will understand the concepts of direct and inverse variation better if real-life eamples are discussed in a qualitative way for problem. Some students are confused by solving for the constant of variation and then using that constant in the original equation and solving for a different variable. 4 4 Answers: a) k =, y =, y =, b) k=6, g=6h, 0 gallons of fuel; a) k =, y =, y =, b) k=50, t= 50, 6 feet s per second; a) k=, r=s t 8y T, b) k=8, =, c) k=, V =, in z P M-8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
8 Additional Eercises 6. Form I Find the domain of each rational epression.. f Name Date.. f.. f( ) Simplify each rational epression y 6y ab a b 6ab Perform each indicated operation and simplify y y m m4 0. m 4 m 5m 6y y. 7z z E-7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
9 Additional Eercises 6. Form II Name Date Find the domain of each rational epression.. f.. f( ) f( ) 67. Simplify each rational epression ab a b 6ab Perform each indicated operation and simplify y y 4y m m m m 5m E-8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
10 Additional Eercises 6. Form III Name Date Find the domain of each rational epression.. f. f 4... f( ) Simplify each rational epression ab a b 6ab Perform each indicated operation and simplify m m4. 4 m 5m m E-9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
11 Additional Eercises 6. Form I Name Date Add or subtract as indicated. Simplify each answer.... z z Find the LCD of the rational epressions in each list. 4., , , Add or subtract as indicated. Simplify each answer E-0 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
12 Additional Eercises 6. Form II Name Date Add or subtract as indicated. Simplify each answer.. y 7y Find the LCD of the rational epressions in each list. 9 4., , , Add or subtract as indicated. Simplify each answer y. y y y.. E- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
13 Additional Eercises 6. Form III Name Date Add or subtract as indicated. Simplify each answer... y Find the LCD of the rational epressions in each list., 8z z , ,, Add or subtract as indicated. Simplify each answer E- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
14 Additional Eercises 6. Form I Name Date Simplify each comple fraction ab b a b b a a a b b a a b b E- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
15 Additional Eercises 6. (cont.) Name a 9. a ab a b E-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
16 Additional Eercises 6. Form II Name Date Simplify each comple fraction ab b a b b a a a b b a a b b E-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
17 Additional Eercises 6. (cont.) Name E-6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
18 Additional Eercises 6. Form III Name Date Simplify each comple fraction yz 6 z y y 5 6 y ab b a b E-7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
19 Additional Eercises 6. (cont.) Name y y E-8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
20 Additional Eercises 6.4 Form I Name Date Divide y 7y y 7y 4 4 y 6 y 6y t t E-9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
21 Additional Eercises 6.4 Form II Name Date Divide. 8 4a b 8a b 6ab y 4 y y y t 4 69t 00 6t t t E-40 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
22 Additional Eercises 6.4 Form III Name Date Divide y y y y 6 4 6a b 0a b 4ab t 5 t t 8 t t t t t E-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
23 Additional Eercises 6.5 Form I Name Date Solve z 7 0z z 0 y 4 y y y E-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
24 Additional Eercises 6.5 Form II Name Date Solve y 4 y y y z. 6z z z 6z 4.. E-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
25 Additional Eercises 6.5 Form III Name Date Solve y y y y 5 y y y4 y E-44 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
26 Additional Eercises 6.6 Form I Solve each equation for the specified variable.. for y. for a a b c Name Date.. Solve.. Find the number that when subtracted from the numerator 4 and subtracted from the denominator of results in a 9. fraction equivalent to If a certain number is subtracted from the numerator and added to the denominator of 7, the new fraction is 9 equivalent to. Find the number. 5. Find the number that when subtracted from the numerator and added to the denominator of 7 results in a fraction equivalent to In the United States, out of 60 homes is heated by wood. At this rate, how many homes in a community of,000 homes are heated by wood? (Source: 005 American Housing Survey for the United States) 7. In a foreign country, out of 0 homes is heated by wood. At this rate, how many homes in a community of,000 homes are heated by wood? 8. In the United States, out of homes is heated by fuel oil. At this rate, how many homes in a community of,000 homes are heated by fuel oil? (Source: 005 American Housing Survey for the United States) 9. Sidney can mow the lawn in 4 hours using her riding mower. Her son can mow the same lawn in hours using a push mower. How long would it take them to mow the lawn working together? 0. Will Wilson takes 6 hours to shoot a movie scene. It takes Steve Nugent twice as long to do the same the job. How long will it take them to shoot the scene, working together? E-45 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
27 Additional Eercises 6.6 (cont.) Name. Pat can wash the family s cars in hours. Pat s son Tom can do the same job in hours. How long will it take them to wash the family s cars working together?. James Lawson takes times as long to go 48 miles upstream as he does to return. If the boat cruises at 0 mph in still water, what is the speed of the current?... John Williams takes 7 9 times as long to go 70 miles. downstream as he does to return. If the boat cruises at 40 mph in still water, what is the speed of the current? E-46 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
28 Additional Eercises 6.6 Form II Solve each equation for the specified variable.. for y y Name Date.. for p m n p. Solve.. Find the number that when added to the numerator and subtracted from the denominator of results in a fraction equivalent to Find the number that when added to the numerator and subtracted from the denominator of results in a fraction equivalent to If a certain number is added to the numerator and subtracted from the denominator of 4, the new fraction is 5. equivalent to. Find the number. 6. In the United States, out of homes is heated by fuel oil. At this rate, how many homes in a community of 4,000 homes are heated by fuel oil? (Source: 005 American Housing Survey for the United States) 7. In a foreign country, out of 0 homes is heated by wood. At this rate, how many homes in a community of 6,000 homes are heated by wood? 8. In the United States, out of 60 homes is heated by wood. At this rate, how many homes in a community of 5,000 homes are heated by wood? (Source: 005 American Housing Survey for the United States) E-47 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
29 Additional Eercises 6.6 (cont.) Name 9. It takes Sam 8 hours to build a section of fence. When David helps him, they can build a section in hours. How long would it take David to build a section working alone? 0. Grant Wilson takes 6 hours to shoot a movie scene. Steven Moss can do the same job in half the time. How long will it take them to shoot the scene, working together?. Juan Ramirez can clean the animal cages at the zoo where he works in 4 hours. His supervisor can do the same job in hours. How long will it take them to clean the animal cages at the zoo if they work together?. Ale Bater takes times as long to go 75 miles upstream as he does to return. If the boat cruises at 0 mph in still water, what is the speed of the current? Lance Greyson takes 5 as long to go 90 miles. downstream as he does to return. If the boat cruises at 4 mph in still water, what is the speed of the current? E-48 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
30 Additional Eercises 6.6 Form III Name Date Solve each equation for the specified variable.. for m m n p.. Solve. for b a b c.. If a certain number is subtracted from the numerator and. subtracted from the denominator of 4, the new fraction is 9 equivalent to. Find the number If a certain number is added to the numerator and subtracted from the denominator of, the new fraction is equivalent to 9. Find the number Find the number that when added to the numerator and added to the denominator of results in a fraction equivalent to In the United States, out of homes is heated by fuel oil. At this rate, how many homes in a community of 48,000 homes are heated by fuel oil? (Source: 005 American Housing Survey for the United States) 7. In a foreign country, out of 0 homes is heated by wood. At this rate, how many homes in a community of 0,000 homes are heated by wood? 8. In the United States, out of 60 homes is heated by wood. At this rate, how many homes in a community of 850 homes are heated by wood? (Source: 005 American Housing Survey for the United States) E-49 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
31 Additional Eercises 6.6 (cont.) Name 9. John Williams takes 7 times as long to go 70 miles upstream as he does to return. If the boat cruises at 40 mph in still water, what is the speed of the current? 0. One model airplane travels 0 kilometers per hour faster than another. The faster plane can travel 400 km in the same time the slower plane travels 50 km. Find the speed of both planes.. Lance Jones takes times as long to go 90 miles upstream as he does to return. If the boat cruises at 4 mph in still water, what is the speed of the current?. Ale Wilson takes 6 hours to shoot a movie scene. Pat Morrison can do the same job in one third the time. How long will it take them to shoot the scene, working together?. Jack can mow the lawn in hours using a push mower. Fran can mow the same lawn in 4 hours using her riding mower. How long would it take them to mow the lawn working together? E-50 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
32 Additional Eercises 6.7 Form I If y varies directly as, find the constant of variation and the direct variation equation for each situation.. y = 6 when = Name Date.. y = when = 6. Solve.. The distance a car travels at a fied rate varies directly as the time. If a car travels 0 miles in hours, how long would it take to travel 600 miles? 4. The volume of a gas at a constant temperature varies inversely as the pressure. Find the volume of a gas under a pressure of 0 pounds if the gas occupies 5 cubic centimeters under a pressure of 0 pounds.. 4. If y varies inversely as, find the constant of variation and the direct variation equation for each situation. 5. y = 0 when = 7 6. y = 0.5 when = Find the constant of variation and the variation equation for each situation. 7. y varies jointly as and the square root of z; y = 0 when = 0, z = a varies directly as the square of b and inversely as the cube of c; a = when b = 4, c = Solve. 9. Hooke s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 0-pound weight attached to the spring stretches the spring 5 inches, find the distance that a 45-pound weight attached to the spring stretches the spring. 0. The period (the time required to complete one swing) of a simple pendulum varies directly as the square root of its length. If a pendulum 6 feet long has a period of seconds, find the period of a pendulum 5 feet long E-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
33 Additional Eercises 6.7 Form II Name Date If y varies directly as, find the constant of variation and the direct variation equation for each situation.. y = 9 when =. y = when = 6.. Solve.. The distance a car travels at a fied rate varies directly as the time. If a car travels 5 miles in 5 hours, how long would it take to travel 780 miles? 4. The volume of a gas at a constant temperature varies inversely as the pressure. Find the volume of a gas under a pressure of 5 pounds if the gas occupies 5 cubic centimeters under a pressure of 0 pounds.. 4. If y varies inversely as, find the constant of variation and the direct variation equation for each situation. 5. y = 0 when = 6. y = 0.8 when = Find the constant of variation and the variation equation for each situation. 7. y varies jointly as and the square root of z; y = 40 when = 0, z = a varies directly as the square of b and inversely as the cube root of c; a = when b = 4, c = Solve. 9. The distance that a freely falling body falls varies directly as the square of the time it falls. If a body falls 400 feet in 5 seconds, how far does it fall in seconds? 0. The period (the time required to complete one swing) of a simple pendulum varies directly as the square root of its length. If a pendulum feet long has a period of 4 seconds, find the period of a pendulum feet long E-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
34 Additional Eercises 6.7 Form III Name Date If y varies directly as, find the constant of variation and the direct variation equation for each situation.. y = 9 when = 7. y = 0.7 when = 4.. Solve.. The time required for a car to travel a fied distance varies inversely as the rate at which it travels. If it takes 5 hours at 64 mph to travel the distance, how long will it takes at 60 mph? 4. The volume of a gas at a constant temperature varies inversely as the pressure. Find the volume of a gas under a pressure of 4 pounds if the gas occupies cubic centimeters under a pressure of 0 pounds.. 4. If y varies inversely as, find the constant of variation and the direct variation equation for each situation. 5. y = 0 when = y = 5 when = 0 6. Find the constant of variation and the variation equation for each situation. 7. y varies jointly as and the square root of z; y = 0 when = 0, z = 6 8. v varies jointly as h and the square of r; v = 07 when h = 8, r = Solve. 9. Hooke s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 0-pound weight attached to the spring stretches the spring inches, find the distance that a 5-pound weight attached to the spring stretches the spring. 0. The period (the time required to complete one swing) of a simple pendulum varies directly as the square root of its length. If a pendulum 0 feet long has a period of seconds, find the period of a pendulum 40 feet long E-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
35 Name: Instructor: Date: Section: Section 6. Rational Functions and Multiplying and Dividing Rational Epressions Objective: Provide practice with multiplying and dividing rational epressions.. Suggested Format: Small Group Time: 0 minutes Find a simplified epression for the area of the figure.. A = lw 9 4. A= bh 9. A = lw Find a simplified epression for the missing side of the rectangle if? A = A- 5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
36 Name: Instructor: Date: Section: Section 6.7 Variation and Problem Solving Objective: Apply concepts of direct and inverse variation. Suggested Format: Small Group Time: 0 minutes The daily sales, S, of Java Jay s Coffee Shop varies directly as their advertising budget, A, and inversely with the price, P, of a cup of coffee. When the advertising budget is $5 per day the coffee shop sells 80 cups of coffee at $.50 each.. Describe what it means for S to vary directly as A.. Describe what it means for S to vary indirectly as P.. Solve for the constant of variation and use this value to write an equation epressing S in terms of A and P. 4. What are the epected daily sales if the price of a cup of coffee goes up to $.75 per cup but the advertising budget remains at $5 per day? 5. What are the epected daily sales if the advertising budget is increased to $0 per day and the price of coffee remains at $.50 per cup? A- 6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
37 Name: Instructor: Date: Section: Chapter 6 Test Form A. Find the domain of the rational epression.. f ( ) = + 0 Write each rational epression in lowest terms.. 6 y 6 0 y Perform the indicated operations. Write answers in lowest terms T- 7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
38 Name: Instructor: Date: Section: Chapter 6 Test Form A cont d Simplify each comple fraction Divide. 5. y + y y Use synthetic division to divide by ( 4) ( ) P = 4 +, use the If ( ) 4 remainder theorem to find P( ). 9. Solve for y. 4y + 6 = 9. T- 8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
39 Name: Instructor: Date: Section: Chapter 6 Test Form A cont d Solve each equation for = 0... = = = The speed of a boat in still water is 0 mph. If 4. the boat travels 40 miles downstream in the same time that it takes to travel 9 miles upstream, find the current of the stream. 5. Suppose that w is directly proportional to y and 5. inversely proportional to v. If w = when v = 0 and y = 0, find w when v = 0 and y = 5. T- 9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
40 Name: Instructor: Date: Section: Chapter 6 Test Form B Find the domain of each rational function Write each rational epression in lowest terms.. 4. y 6 y 48y 6 y Perform the indicated operation. Write answers in lowest terms y 4yz y z 8 y 6. a + b a b a b a + ab p + p p p p p T- 0 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
41 Name: Instructor: Date: Section: Chapter 6 Test Form B cont d Divide... y y + y 6y Use synthetic division to divide ( + + ) by ( ) P = + 7 +, find P() If ( ) 4 Simplify the comple fractions Solve each equation for = = =. T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
42 Name: Instructor: Date: Section: Chapter 6 Test Form B cont d. = If a number is subtracted from the numerator. of 5 and the same number is added to the denominator of, the result is equivalent to. 5 Find the number. 4. Suppose that w is proportional to the square root 4. of v. If w is 0 when v is 6, find w when v is Suppose that b is directly proportional to c, and 5. inversely proportional to d. If b = 6 when c = and d = 6, find b when c = 8 and d = 8. T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
43 Name: Instructor: Date: Section: Chapter 6 Test Form C. Find the domain of the rational function.. 7 f ( ) = 4 Write each rational epression in lowest terms... 00a bc 4 40a c y 4y Perform the indicated operation. Write answers in lowest terms a + a. +. a a T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
44 Name: Instructor: Date: Section: Chapter 6 Test Form C cont d Simplify each comple fraction Divide a b + 45a b 0ab 5ab ( 7 5) ( ) Use synthetic division to divide ( ) ( ) 8. If P( ) = + 5, find ( ) Solve each equation for. P = = = T- 4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
45 Name: Instructor: Date: Section: Chapter 6 Test Form C cont d. a =. a 4. Three workers can complete a job in hours,. 5 hours, and 6 hours respectively. How long will it the workers to complete the job if they work together? 4. One number is four times another. The sum of 4. their reciprocals is. Find the two numbers Suppose that b is proportional to c, and inversely 5. proportional to the square of d. If b = 45 when c = 40 and d =, find b when c = 0 and d =. T- 5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
46 Name: Instructor: Date: Section: Chapter 6 Test Form D Circle the correct answer.. Find the domain of the rational function f ( ) = { is a real number and,, 4} b. { is a real number and, } { is a real number and, 4, 6} d. { is a real number and, 6} Write each rational epression in lowest terms y y z 5 z b. yz 4z d. 4yz. ( ) b d Perform the indicated operations. Write answers in lowest terms y 8 6 y y 9y y b. 5 8 y d. 5 6 y ( 4) b. 5( + 4) 5 ( 4) d. 0 T- 6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
47 Name: Instructor: Date: Section: Chapter 6 Test Form D cont d ( ) ( 5)( + 4) b. ( ) ( 5)( + ) ( + 4) ( 5)( 4) d y y y y y 6y ( + ) b. ( y) d. ( ) ( ) y 8. ( ) b. 8 + d ( 4 )( + 4 ) b d. ( + 4 )( 4 ) b d ( 4 )( + 6 )( ) 9 ( 4 )( + 6 )( ) b. ( 4 )( + 6 ) 9 d. ( 4 )( + 6 )( ) T- 7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
48 Name: Instructor: Date: Section: Chapter 6 Test Form D cont d y 6y y 4 + b. y 4 + y y y 6 + d. 4y + 8. ( ) ( ) 7 + b. + d Use synthetic division to divide. + + by b d P = Use the remainder theorem to find P( ) for ( ) 4 46 b d. 8 Simplify each comple fraction b. 4( 6) d. 6 7 T- 8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
49 Name: Instructor: Date: Section: Chapter 6 Test Form D cont d b. + + d b. d. Solve each equation for = 5 b. 6 d = 0, b. d. 5. = b., 6 d.. + = + 4, b., d. T- 9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
50 Name: Instructor: Date: Section: Chapter 6 Test Form D cont d. Solve for a + b a = b 6 5 a = b 8 b. a = b b + a = d. 8 b a = 8 4. Suppose that Q is jointly proportional to R and S. If Q = 0 when R = 6 and S = 5, find Q when R = 9 and S =. 66 b. 99 d. 5. The sum of the reciprocals of two consecutive integers is 5. Find the integers. 56, 4 b., 6, 7 d., 4 T- 0 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
51 Name: Instructor: Date: Section: Chapter 6 Test Form E Circle the correct answer. 4. Find all numbers for which the rational epressions 4 + is undefined. =,,, 6 b. = 0,, 6 = 0,, 4, 6 d. = 4. Solve y + 6 = for y. y y = b. y = 6 y = d. y + 6 y = Write each rational epression in lowest terms yz y z 4 y z b. 4 z 4y 4y d. z y z b. ( + )( + 4 ) + 8 d. + 8 Perform the indicated operations. Write answers in lowest terms y 9y 5y 5y y b. y 9 y 5 d. y T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
52 Name: Instructor: Date: Section: Chapter 6 Test Form E cont d y y y y y y y b. d. y y ( ) ( ) + b. ( + ) ( )( ) d. ( + )( ) b. + d b d ( + 5 )( 6 ) b. ( + 5 )( 6 )( ) + d. ( + 5 )( 6 )( ) b d. 0 + T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
53 Name: Instructor: Date: Section: Chapter 6 Test Form E cont d Simplify each comple fraction b. d ( + )( ) b. ( + )( + 9 ) 6 d b d Divide. 5. y y + y 4y y b. y + y y d. y b d T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
54 Name: Instructor: Date: Section: Chapter 6 Test Form E cont d 7. Use synthetic division to divide by b d P = use the remainder theorem to find P( ). 8. If ( ) 4 8 b. 8 4 d. Solve each equation for = b. 0 5 d = b. d = b. 8, d = + 9 +, 7 b., 7 7, d. 7,. Solve for + = a 6 a = + 4 b. 6 + a = a = d. a = 6 T- 4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
55 Name: Instructor: Date: Section: Chapter 6 Test Form E cont d 4. A plane flies 75 miles with the wind and 95 miles against the wind in the same amount of time. If the speed of the plane in still air is 50 mph, find the speed of the wind. 45 mph b. 65 mph 55 mph d. 60 mph 5. Suppose that B is proportional to C and inversely proportional to D. If B = 5 when C = 60 and D =, find B when C = 40 and D = 4. 5 b. 0 0 d. 5 T- 5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
56 Name: Instructor: Date: Section: Chapter 6 Test Form F Circle the correct answer. Write each rational epression in lowest terms b. ( ) + d. ( ) y 4y y y y b y d. y Find the domain of the rational function.. f ( ) = { is a real number and 9,, 0} b. { is a real number and 9, } { is a real number and, 0, 9} d. { is a real number and, 9} 4. Solve for a c a = c + d a d c = b. c = ad c = d d. a a ad c = a + Perform the indicated operations. Write answers in lowest terms ( ) b. 4 5 ( + ) d. 4 5 ( + ) T- 6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
57 Name: Instructor: Date: Section: Chapter 6 Test Form F cont d y y y ( ) b. d ( + ) 6 5 b. ( ) 6 4 ( ) ( ) 4 d. 4 6 ( ) ( 4 )( + 6 ) b. ( 4 )( + 6 ) + 5 ( 4 )( + 6 ) 5 + d. ( 4 )( + 6 ) b. ( + ) ( ) d. 8 Divide. 0. a b a b ab 5ab a b ab b. 4a b ab ab d. 0a b 0a b b d T- 7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
58 Name: Instructor: Date: Section: Chapter 6 Test Form F cont d. Use synthetic division to divide by b d If ( ) P = , use the remainder theorem to find P. 5 b. 5 d. Simplify each comple fraction b. 0 + d b d b. + d. 7 T- 8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
59 Name: Instructor: Date: Section: Chapter 6 Test Form F cont d b d Add: b d Solve each equation for = 6, b. 5 6, 5 5, d. 6 6,0, = b. 8 d. 6, 4 + = , b. 4,7 7 d. 8, 9. = + 8, 6 b. 6, 9 d. 9 T- 9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
60 Name: Instructor: Date: Section: Chapter 6 Test Form F cont d. One number is 5 times another. The sum of their reciprocals is the fraction 5. Find the two numbers. 5, 6 b. 6, 0, 0 6 d., 4 4. Two trains leave at the same time going in opposite directions. One train travels 5 mph faster than the other. Five hours later, the trains are 675 miles apart. Find the speed of each. 50 mph, 75 mph b. 45 mph, 70 mph 55 mph, 80 mph d. 65 mph, 90 mph 5. Suppose B is proportional to the square of C and inversely proportional to D. If B = 7 when C = 6 and D = 6, find B when C = and D = 6. 8 b. 4 d. 6 T- 40 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall
Unit 3: Rational Expressions
Unit : Rational Epressions Common Denominators Directions: For each of the following, practice finding the LCM necessary for creating a common denominator (Hint: make sure to factor). 1) ; 0. 14; 1 10
More informationAdditional Exercises 8.7 Form I Applications Using Rational Equations and Variation
Additional Exercises 8.7 Form I Applications Using Rational Equations and Variation 1. A cyclist bikes at a constant speed for 20 miles. He then returns 1. home at the same speed but takes a different
More informationINTERMEDIATE ALGEBRA REVIEW FOR TEST 3
INTERMEDIATE ALGEBRA REVIEW FOR TEST 3 Evaluate the epression. ) a) 73 (-4)2-44 d) 4-3 e) (-)0 f) -90 g) 23 2-4 h) (-2)4 80 i) (-2)5 (-2)-7 j) 5-6 k) 3-2 l) 5-2 Simplify the epression. Write your answer
More information7.1 Rational Expressions and Their Simplification
7.1 Rational Epressions and Their Simplification Learning Objectives: 1. Find numbers for which a rational epression is undefined.. Simplify rational epressions. Eamples of rational epressions: 3 and 1
More informationMath 102, Intermediate Algebra Author: Debra Griffin Circle one: Red Bluff Main Campus Burney Weaverville
Math 02, Intermediate Algebra Name Author: Debra Griffin Circle one: Red Bluff Main Campus Burney Weaverville Chapter 2 Rational Epressions 2. Rational Epressions and Allowed X-Values (Domain) 2 Determining
More informationMini-Lecture 5.1 Exponents and Scientific Notation
Mini-Lecture.1 Eponents and Scientific Notation Learning Objectives: 1. Use the product rule for eponents.. Evaluate epressions raised to the zero power.. Use the quotient rule for eponents.. Evaluate
More information6.7 Variation and Problem Solving. OBJECTIVES 1 Solve Problems Involving Direct Variation. 2 Solve Problems Involving Inverse Variation.
390 CHAPTER 6 Rational Epressions 66. A doctor recorded a body-mass inde of 7 on a patient s chart. Later, a nurse notices that the doctor recorded the patient s weight as 0 pounds but neglected to record
More informationMini-Lecture 8.1 Solving Quadratic Equations by Completing the Square
Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.
More informationMATH98 Intermediate Algebra Practice Test Form B
MATH98 Intermediate Algebra Practice Test Form B MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y - 4) - (y + 9) = y 1) -
More informationAlgebra 2 Chapter 9 Page 1
Section 9.1A Introduction to Rational Functions Work Together How many pounds of peanuts do you think and average person consumed last year? Us the table at the right. What was the average peanut consumption
More informationDue for this week. Slide 2. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
MTH 09 Week 3 Due for this week Homework 3 (on MyMathLab via the Materials Link) The fifth night after class at 11:59pm. Read Chapter 6.6, 8.4 and 11.1-11.5 Do the MyMathLab Self-Check for week 3. Learning
More informationMini-Lecture 7.1 Radicals and Radical Functions
Mini-Lecture 7. Radicals and Radical Functions Learning Objectives:. Find square roots.. Approimate roots.. Find cube roots.. Find n th roots.. Find n a n when a is an real number. 6. Graph square and
More informationFLC Ch 6. Simplify. State any restrictions (if necessary). a) b) Simplify each. List all restrictions on the domain. Next, graph the function f.
Math 0 Intermediate Algebra Defn Sec 6.: Rational Expressions and Functions: Multiplying and Dividing A polynomial divided by another nonzero polynomial is called a rational expression. If P and Q are
More informationHonors Algebra 2 Chapter 9 Page 1
Introduction to Rational Functions Work Together How many pounds of peanuts do you think and average person consumed last year? Us the table at the right. What was the average peanut consumption per person
More informationWhy? 2 3 times a week. daily equals + 8_. Thus, _ 38 or 38% eat takeout more than once a week. c + _ b c = _ a + b. Factor the numerator. 1B.
Then You added and subtracted polynomials. (Lesson 7-5) Now Add and subtract rational epressions with like denominators. 2Add and subtract rational epressions with unlike denominators. Adding and Subtracting
More informationAssignment #1 MAT121 Summer 2015 NAME:
Assignment #1 MAT11 Summer 015 NAME: Directions: Do ALL of your work on THIS handout in the space provided! Circle your final answer! On problems that your teacher would show work on be sure that you also
More informationName. Unit 1 Worksheets Math 150 College Algebra and Trig
Name Unit 1 Worksheets Math 10 College Algebra and Trig Revised: Fall 009 Worksheet 1: Integral Eponents Simplify each epression. Write all answers in eponential form. 1. (8 ). ( y). (a b ). y 6. (7 8
More informationMAT 1033 Final Review for Intermediate Algebra (Revised April 2013)
1 This review corresponds to the Charles McKeague textbook. Answers will be posted separately. Section 2.1: Solve a Linear Equation in One Variable 1. Solve: " = " 2. Solve: "# = " 3. Solve: " " = " Section
More informationBasic Property: of Rational Expressions. Multiplication and Division of Rational Expressions. The Domain of a Rational Function: P Q WARNING:
Basic roperties of Rational Epressions A rational epression is any epression of the form Q where and Q are polynomials and Q 0. In the following properties, no denominator is allowed to be zero. The Domain
More informationMy Math Plan Assessment #2 Study Guide
My Math Plan Assessment #2 Study Guide 1. Find the x-intercept and the y-intercept of the linear equation. 8x y = 4 2. Use factoring to solve the quadratic equation. x 2 + 9x + 1 = 17. Multiply and simplify
More informationMATH 110: FINAL EXAM REVIEW
MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = 7 + 8 ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set
More informationFor problems 1 4, evaluate each expression, if possible. Write answers as integers or simplified fractions
/ MATH 05 TEST REVIEW SHEET TO THE STUDENT: This Review Sheet gives you an outline of the topics covered on Test as well as practice problems. Answers are at the end of the Review Sheet. I. EXPRESSIONS
More informationCHAPTER 2 Solving Equations and Inequalities
CHAPTER Solving Equations and Inequalities Section. Linear Equations and Problem Solving........... 8 Section. Solving Equations Graphically............... 89 Section. Comple Numbers......................
More informationCHAPTER 11: RATIONAL EQUATIONS AND APPLICATIONS
CHAPTER 11: RATIONAL EQUATIONS AND APPLICATIONS Chapter Objectives By the end of this chapter, students should be able to: Identify extraneous values Apply methods of solving rational equations to solve
More information4.7 Solutions of Rational Equations
www.ck1.org Chapter 4. Rational Equations and Functions 4.7 s of Rational Equations Learning Objectives Solve rational equations using cross products. Solve rational equations using lowest common denominators.
More informationRational Equations. You can use a rational function to model the intensity of sound.
UNIT Rational Equations You can use a rational function to model the intensit of sound. Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations,
More informationSolve. Label any contradictions or identities. 1) -4x + 2(3x - 3) = 5-9x. 2) 7x - (3x - 1) = 2. 3) 2x 5 - x 3 = 2 4) 15. 5) -4.2q =
Spring 2011 Name Math 115 Elementary Algebra Review Wednesday, June 1, 2011 All problems must me done on 8.5" x 11" lined paper. Solve. Label any contradictions or identities. 1) -4x + 2(3x - 3) = 5-9x
More informationC. 3 PRACTICE FINAL EXAM. 1. Simplify B. 2. D. E. None of the above. 2. Factor. completely. E. None of the above. 3.
MA 1500 1. Simplify ; A. B. 2 C. 2. Factor 8 completely. A. x y x + y B. x 2y C. 2x y 2x + y 2x y. Simplify ( ) ; A. a b c B. C. a b c a b 6c c a b 1 MA 1500 4. Subtract and simplify. + 2 A. x: x ; x;
More informationUnit 8 Rational Expressions and Equations Examples Introductory Algebra Page 1 of 18
Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8 When canceling quantities like a, to avoid the division by zero case we usually specify that a 0. a Remember that you can
More information, passing through the point 4, 2
MATD 090 INTERMEDIATE ALGEBRA Review for Test Test covers all cumulative material, including new sections.-.8,.-.,.6-.7, and 6.-6.. Bring a non-graphing calculator and something to write with and erase
More informationReady To Go On? Skills Intervention 12-1 Inverse Variation
12A Find this vocabular word in Lesson 12-1 and the Multilingual Glossar. Identifing Inverse Variation Tell whether the relationship is an inverse variation. Eplain. A. Read To Go On? Skills Intervention
More informationChapter 10. Section 10.1 = 2(27) Practice Problems. The value of 2y + y 6 when y = 3 is 57. The height of the object at 1 second is 514 feet.
Chapter 0 Section 0. Practice Problems. (+ + ( ( + ( ( + (. ( + ( + 0. ( z.z+ + (z.z+. z + z.z.z+ +. z.z+.. +. +... ( + ( + ( + ( ( + ( ( +. (b+ (b 0 (b+ + ( b+ 0 b b+ + 0 b +. ( + + ( + ( + + + ( + +
More informationBasic Property: of Rational Expressions
Basic Properties of Rational Expressions A rational expression is any expression of the form P Q where P and Q are polynomials and Q 0. In the following properties, no denominator is allowed to be zero.
More informationMINI-LECTURES AND ANSWERS
MINI-LECTURES AND ANSWERS Mini-Lecture R. INTERMEDIATE ALGEBRA. Write sets using set notation.. Use number lines.. Know the common sets of numbers. 4. Find additive inverses. 5. Use absolute value. 6.
More informationCHAPTER 5 RATIONAL FUNCTIONS
CHAPTER 5 RATIONAL FUNCTIONS Big IDEAS: ) Graphing rational functions ) Performing operations with rational epressions 3) Solving rational equations Section: 5- Model Inverse and Joint Variation Essential
More informationQUESTIONS 1-46 REVIEW THE OBJECTIVES OF CHAPTER 2.
MAT 101 Course Review Questions Valid for Fall 2014, Spring 2015 and Summer 2015 MIDTERM EXAM FINAL EXAM Questions 1-86 are covered on the Midterm. There are 25 questions on the midterm, all multiple choice,
More informationName Class Date. You can use the properties of equality to solve equations. Subtraction is the inverse of addition.
2-1 Reteaching Solving One-Step Equations You can use the properties of equality to solve equations. Subtraction is the inverse of addition. What is the solution of + 5 =? In the equation, + 5 =, 5 is
More informationMATH 103 Sample Final Exam Review
MATH 0 Sample Final Eam Review This review is a collection of sample questions used b instructors of this course at Missouri State Universit. It contains a sampling of problems representing the material
More informationSection 7.1 Rational Functions and Simplifying Rational Expressions
Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.1 Section 7.1 Rational Functions and Simplifying Rational Expressions Complete the outline as you view Video Lecture 7.1. Pause the video
More informationMATH ALGEBRA AND FUNCTIONS
Students: 1. Students express quantitative relationships using algebraic terminology, expressions, equations, inequalities and their graphs. 1. Use variables and appropriate operations to write an expression,
More information5.1 The Language of Mathematics
5. The Language of Mathematics Prescribed Learning Outcomes (PLO s): Use mathematical terminology (variables, degree, number of terms, coefficients, constant terms) to describe polynomials. Identify different
More informationMy Math Plan Assessment #3 Study Guide
My Math Plan Assessment # Study Guide 1. Identify the vertex of the parabola with the given equation. f(x) = (x 5) 2 7 2. Find the value of the function. Find f( 6) for f(x) = 2x + 11. Graph the linear
More informationMATH98 Intermediate Algebra Practice Test Form A
MATH98 Intermediate Algebra Practice Test Form A MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y - 4) - (y + ) = 3y 1) A)
More informationChapter 4 Polynomial and Rational Functions
Chapter Polynomial and Rational Functions - Polynomial Functions Pages 09 0 Check for Understanding. A zero is the value of the variable for which a polynomial function in one variable equals zero. A root
More informationIntermediate Algebra 100A Final Exam Review Fall 2007
1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,
More information2. Which of the following expressions represents the product of four less than three times x and two more than x?
Algebra Topics COMPASS Review You will be allowed to use a calculator on the COMPASS test. Acceptable calculators are: basic calculators, scientific calculators, and graphing calculators up through the
More informationFinal Exam Review Part 1 #5
Final Exam Review Part #5 Intermediate Algebra / MAT 35 Fall 206 Master (Prof. Fleischner) Student Name/ID:. Solve the compound inequality. 4x + 2 0 and 3x 4 < 8 Graph the solution on the number line.
More informationLESSON #34 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II
LESSON #4 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II A rational epression is a fraction that contains variables. A variable is very useful in mathematics. In
More information150. a. Clear fractions in the following equation and write in. b. For the equation you wrote in part (a), compute. The Quadratic Formula
75 CHAPTER Quadratic Equations and Functions Preview Eercises Eercises 8 50 will help you prepare for the material covered in the net section. 8. a. Solve by factoring: 8 + - 0. b. The quadratic equation
More informationFinal Exam Review Part 1 #4
Final Exam Review Part #4 Intermediate Algebra / MAT 35 Fall 206 Master (Prof. Fleischner) Student Name/ID:. Solve the compound inequality. 5 < 2x 3 3 Graph the solution on the number line. - -0-9 -8-7
More information1 Linear and Absolute Value Equations
1 Linear and Absolute Value Equations 1. Solve the equation 11x + 6 = 7x + 15. Solution: Use properties of equality to bring the x s to one side and the numbers to the other: 11x (7x) + 6 = 7x (7x) + 15
More informationAppendix A A318. Appendix A.1 (page A8) Vocabulary Check (page A8) Answers to All Exercises and Tests. x (c) Bounded
A Answers to All Eercises and Tests Appendi A Appendi A. (page A) Vocabulary Check (page A). rational. irrational. absolute value. composite. prime. variables; constants. terms. coefficient 9. Zero-Factor
More informationMini Lecture 9.1 Finding Roots
Mini Lecture 9. Finding Roots. Find square roots.. Evaluate models containing square roots.. Use a calculator to find decimal approimations for irrational square roots. 4. Find higher roots. Evaluat. a.
More informationMini-Lecture 2.1 Simplifying Algebraic Expressions
Copyright 01 Pearson Education, Inc. Mini-Lecture.1 Simplifying Algebraic Expressions 1. Identify terms, like terms, and unlike terms.. Combine like terms.. Use the distributive property to remove parentheses.
More informationCHAPTER 1 Equations, Inequalities, and Mathematical Modeling
CHAPTER Equations, Inequalities, and Mathematical Modeling Section. Graphs of Equations.................... 9 Section. Linear Equations in One Variable............. Section. Modeling with Linear Equations..............
More informationHAlgebra 2: Unit 7: Chapter 9 Spring Day Date Lesson Assignment. Section 9.1: Direct, Inverse, & Joint Variation Classwork: Packet p.
HAlgebra : Unit 7: Chapter Spring 0 Name RATIONAL FUNCTIONS. NC Objectives:.0 Operate with algebraic epressions (polynomial, rational, comple fractions) to solve problems..0 Model and solve problems using
More informationLecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 3rd edition. Miller, O'Neill, & Hyde. Victor Valley College
Lecture Guide Math 90 - Intermediate Algebra to accompany Intermediate Algebra, 3rd edition Miller, O'Neill, & Hyde Prepared by Stephen Toner Victor Valley College Last updated: 7/21/14 6.1 Introduction
More information1. complex; complex 2. 18; 12; 30; 9; 4; 5; ; 4. 2x; 3; 5x; Copyright 2013 Pearson Education, Inc.
Chapter 7: Rational Epressions 7.5 Chec Points... 4 4 Add to get a single rational epression in the numerator. 8 4 Subtract to get a single rational epression in the denominator. 8 5 4 Perform the division
More informationAdvanced Algebra Scope and Sequence First Semester. Second Semester
Last update: April 03 Advanced Algebra Scope and Sequence 03-4 First Semester Unit Name Unit : Review of Basic Concepts and Polynomials Unit : Rational and Radical Epressions Sections in Book 0308 SLOs
More informationMath 100 Exam 4 Review (Chapters 7-8) Name
Math 100 Eam 4 Review (Chapters 7-8) Name The problems included in this review involve the important concepts covered in chapters 7 and 8. Be prepared to write soltuions on the board if called upon. Find
More information2-2. Warm Up. Simplify each expression. 1. ( 7)(2.8) ( 9)( 9)
Warm Up Simplify each expression. 1. ( 7)(2.8) 2. 0.96 6 3. ( 9)( 9) 4. 5. 6. Learning Goals 1. Students will solve and check equations using multiplication 2. Students will solve and check equations using
More informationChapter 6 RATIONAL EXPRESSIONS AND EQUATIONS
Name: Instructor: Date: Section: Chapter 6 RATIONAL EXPRESSIONS AND EQUATIONS 6.1 Multiplying and Simplifying Rational Expressions Learning Objectives a Find all numbers for which a rational expression
More information8-1 Study Guide and Intervention
8-1 Study Guide and Intervention Simplify Rational Epressions A ratio of two polynomial epressions is a rational epression. To simplify a rational epression, divide both the numerator and the denominator
More informationSection 5.1 Model Inverse and Joint Variation
108 Section 5.1 Model Inverse and Joint Variation Remember a Direct Variation Equation y k has a y-intercept of (0, 0). Different Types of Variation Relationship Equation a) y varies directly with. y k
More informationAlgebra. Robert Taggart
Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit : Algebra Basics Lesson : Negative and Positive Numbers....................... Lesson : Operations
More information7.6 Radical Equations and Problem Solving
Section 7.6 Radical Equations and Problem Solving 447 Use rational eponents to write each as a single radical epression. 9. 2 4 # 2 20. 25 # 2 3 2 Simplify. 2. 240 22. 2 4 6 7 y 0 23. 2 3 54 4 24. 2 5-64b
More informationPrecision and Accuracy. Learning Targets: Unit 2.1 To determine the degree of precision of a measurement.
Precision and Accuracy Learning Targets: Unit.1 To determine the degree of precision of a measurement. We often use numbers that are not exact. Measurements are approximate there is no such thing as a
More informationReview for Intermediate Algebra (MATD 0390) Final Exam Oct 2009
Review for Intermediate Algebra (MATD 090) Final Eam Oct 009 Students are epected to know all relevant formulas, including: All special factoring formulas Equation of a circle All formulas for linear equations
More informationCCR Math - Grade 7 Practice Test
R Math - Grade 7 Practice Test You may use a calculator for questions -7.. Use the picture below to answer the question. A B What is the probability of spinning a? A. B.. D. 5 3 5 3 5 A 3 Go on to the
More informationMultiplying and Dividing Rational Expressions y y v 2 3 v 2-13v x z 25 x. n - 6 n 2-6n. 6x + 2 x 2. w y a 3 w.
8- Multiplying and Dividing Rational Epressions Simplify each epression.. 9 a b 7 a b c. ( m n ) -8 m 5 n. 0 y + 5y 5 y - 5y. k - k - 5 k - 9 5. 5 - v v - v - 0. + - - 7. - u y 5 z 5 5 u y 8. a + y y +
More informationASU Mathematics Placement Test Sample Problems June, 2000
ASU Mathematics Placement Test Sample Problems June, 000. Evaluate (.5)(0.06). Evaluate (.06) (0.08). Evaluate ( ) 5. Evaluate [ 8 + ( 9) ] 5. Evaluate 7 + ( ) 6. Evaluate ( 8) 7. Evaluate 5 ( 8. Evaluate
More informationAlgebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
More informationSolving Multiplication Equations. Each friend pays $2. The solution of 3x = 6 is 2. Solve each equation using models or a drawing.
3-3 Solving Multiplication Equations MAIN IDEA Solve multiplication equations. New Vocabulary formula Math Online glencoe.com Concepts In Motion Etra Eamples Personal Tutor Self-Check Quiz MONEY Suppose
More informationCHAPTER 8 Quadratic Equations, Functions, and Inequalities
CHAPTER Quadratic Equations, Functions, and Inequalities Section. Solving Quadratic Equations: Factoring and Special Forms..................... 7 Section. Completing the Square................... 9 Section.
More informationPolynomial and Rational Functions
Polynomial and Rational Functions Figure 1 5-mm film, once the standard for capturing photographic images, has been made largely obsolete by digital photography. (credit film : modification of work by
More informationAbout the Portfolio Activities. About the Chapter Project
Galileo is credited as the first person to notice that the motion of a pendulum depends only upon its length. About the Chapter Project Finding an average is something that most people can do almost instinctively.
More informationMath Analysis Chapter 2 Notes: Polynomial and Rational Functions
Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do
More information3. A tennis field has length 78 feet and width of 12 yards. What is the area of the field (in square feet)?
Station 1: MSG9-12.A1.NQ.1: Use units of measure (linear, area, capacity, rates, and time) as a way to understand problems; identify, use and record appr opriate units of measure within context, within
More informationSolution: Slide 7.1-3
7.1 Rational Expressions and Functions; Multiplying and Dividing Objectives 1 Define rational expressions. 2 Define rational functions and describe their domains. Define rational expressions. A rational
More informationQuiz # 6 Date: Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Quiz # 6 Date: Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all numbers not in the domain of the function. ) f(x) = x + None - 0 ) )
More information6. y = 4_. 8. D: x > 0; R: y > 0; 9. x 1 (3)(12) = (9) y 2 36 = 9 y 2. 9 = _ 9 y = y x 1 (12)(60) = x 2.
INVERSE VARIATION, PAGES 676 CHECK IT OUT! PAGES 686 a. No; the product y is not constant. b. Yes; the product y is constant. c. No; the equation cannot be written in the form y k. y 5. D: > ; R: y > ;
More informationMath 060/Final Exam Review Guide/ / College of the Canyons
Math 060/Final Exam Review Guide/ 010-011/ College of the Canyons General Information: The final exam is a -hour timed exam. There will be approximately 40 questions. There will be no calculators or notes
More informationA2T. Rational Expressions/Equations. Name: Teacher: Pd:
AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest
More informationMA 22000, Lesson 2 Functions & Addition/Subtraction Polynomials Algebra section of text: Sections 3.5 and 5.2, Calculus section of text: Section R.
MA 000, Lesson Functions & Addition/Subtraction Polynomials Algebra section of tet: Sections.5 and 5., Calculus section of tet: Section R.1 Definition: A relation is any set of ordered pairs. The set of
More informationHonors and Regular Algebra II & Trigonometry Summer Packet
Honors and Regular Algebra II & Trigonometry Summer Packet Hello Students, Parents, and Guardians! I hope everyone is enjoying the summer months and the time to rela and recoup. To add to your summer fun,
More informationPre-Algebra 2. Unit 8. Rational Equations Name Period
Pre-Algebra Unit 8 Rational Equations Name Period Basic Skills (7B after test) Practice PAP Algebra Name Per NON-CALCULATOR Simplify: 1. 1 1. 3 5 1 3. 5 9 7 4 1 4. 8 5 9 1 1 1 1 4 6 5. 1 3 5 7 3 6. 5
More informationMini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models
Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic epressions.. Translate English phrases into algebraic epressions.. Determine whether a number is a solution
More informationAlgebra I End of Course Review
Algebra I End of Course Review Properties and PEMDAS 1. Name the property shown: a + b + c = b + a + c (Unit 1) 1. Name the property shown: a(b) = b(a). Name the property shown: m + 0 = m. Name the property
More informationAlgebra Workbook WALCH PUBLISHING
Algebra Workbook WALCH PUBLISHING Table of Contents Unit 1: Algebra Basics Activity 1 What Are Negative and Positive Numbers? I...1 Activity 2 What Are Negative and Positive Numbers? II..2 Activity 3 Larger
More informationLesson 9.1 Skills Practice
Lesson 9.1 Skills Practice Name Date Call to Order Inequalities Vocabulary Write the term that best completes each statement. 1. A(n) in one variable is the set of all points on a number line that makes
More informationCopyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Chapter : Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra. + + 0 The solution set is [0, ).. () The solution set is [, ). 0. >.. > >. The solution set is (., ).. The solution set
More information1.2 Constructing Models to Solve Problems
1.2 Constructing Models to Solve Problems In the previous section, we learned how to solve linear equations. In this section, we will put those skills to use in order to solve a variety of application
More information2.1 The Rectangular Coordinate System
. The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table
More informationMATH ALGEBRA AND FUNCTIONS
Students: 1. Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations and graph and interpret their results.
More informationChapter 1 Equations and Inequalities
Chapter Equations and Inequalities Section.. +. +. ( ) ( ) +. (r ) + (r ) r + r r r r. + + +. +.. +........ +.().( + ) +. +...... ( + )( ) (+ )( ) + ( ) + + +. [ ( )] ( + ) ( + ) + +. + ( ) + Conditional
More informationMultiplication and Division
UNIT 3 Multiplication and Division Skaters work as a pair to put on quite a show. Multiplication and division work as a pair to solve many types of problems. 82 UNIT 3 MULTIPLICATION AND DIVISION Isaac
More informationAlgebraic Expressions and Identities
9 Algebraic Epressions and Identities introduction In previous classes, you have studied the fundamental concepts of algebra, algebraic epressions and their addition and subtraction. In this chapter, we
More informationPRE-ALGEBRA SUMMARY WHOLE NUMBERS
PRE-ALGEBRA SUMMARY WHOLE NUMBERS Introduction to Whole Numbers and Place Value Digits Digits are the basic symbols of the system 0,,,, 4,, 6, 7, 8, and 9 are digits Place Value The value of a digit in
More informationAlgebra II Notes Rational Functions Unit Rational Functions. Math Background
Algebra II Notes Rational Functions Unit 6. 6.6 Rational Functions Math Background Previously, you Simplified linear, quadratic, radical and polynomial functions Performed arithmetic operations with linear,
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More information