Section 1: Expressions Section 1 Topic 1 Using Expressions to Represent Real-World Situations

Transcription

1 Section 1: Expressions Section 1 Topic 1 Using Expressions to Represent Real-World Situations The Florida Turnpike is a 320-mile stretch of highway running from Central Florida to South Florida. Drivers who use a SunPass pay discounted toll rates. For 2-axle passenger vehicles with a SunPass, the toll at the Orlando (I-4) Mile Post is $0.53, and the toll at the Okeechobee Plaza exit in West Palm Beach is $1.04. A 2-axle passenger vehicle with a SunPass drives through the Orlando (I-4) Mile Post five times. Determine the total amount of money that will be deducted from this vehicle s SunPass account. $ $2. 65 Create an algebraic expression to describe the total amount of money that will be deducted from this vehicle s SunPass account for any given number of drives through the Orlando (I-4) Mile Post. Let x be the number of drives through I-4 Mile Post. $0. 53 x 0. 53x

2 The same 2-axle passenger vehicle drives through the Okeechobee Plaza exit in West Palm Beach seven times. Determine the total amount of money that will be deducted from its SunPass account. $ $7. 28 Create an algebraic expression to describe the total amount of money that will be deducted from this vehicle s SunPass account for any given number of drives through the Okeechobee Plaza exit. Let y be the number of drives through Okeechobee. $1. 04 y 1. 04y Write an algebraic expression to describe the combined total amount of money that will be deducted from this vehicle s SunPass account after it passes through the Orlando (I-4) Mile Post and the Okeechobee Plaza exit any given number of times. Remember that x is the number of drives through I-4 Mile Post and y is the number of drives through Okeechobee 0. 53x y What is the total amount of money that will be deducted from this vehicle s SunPass account after it passes through the Orlando (I-4) Mile Post eight times and Okeechobee Plaza exit four times? 0. 53x y = = The total amount is $8. 40.

3 Let s Practice! 1. Mario and Luigi plan to buy a Nintendo Switch for $ Nintendo Switch games cost $59.99 each. They plan to purchase one console. a. Use an algebraic expression to describe how much they will spend, before sales tax, based on purchasing the console and the number of games. Let g represent the number of games c g b. If they purchase one console and three games, how much will they spend before sales tax? = c. Mario and Luigi want to purchase some extra controllers for their friends. Each controller costs $ Use an algebraic expression to describe how much they will spend in total, before sales tax, based on purchasing the console, the number of games, and the number of extra controllers. Let r represent the number of controllers c g r d. What will be the total cost, before sales tax, if Mario and Luigi purchase one console, three games, and two extra controllers? =

4 When defining variables, choose variables that make sense to you, such as h for hours and d for days. Try It! 2. The Thomas family likes to visit Everglades National Park, the largest tropical wilderness in the U.S., which partially covers the Miami-Dade, Monroe, and Collier counties in Florida. They hold an annual pass that costs $ Each canoe rental costs $26.00 and each bicycle rental costs $ Use an algebraic expression to describe how much they spend in a year based on the annual pass, the number of canoe rentals, and the number of bicycle rentals. Identify the parts of the expression by underlining the coefficient(s), circling the constant(s), and drawing a box around the variable(s). Let c be the number of canoe rentals and b the number of bicycle rentals. Then, c + 15b c + 15b 3. Ramiro brought home ten pounds of rice to add to the 24 ounces of rice he had in the pantry. Let x represent the amount of rice Ramiro uses over the next few days. a. Write an algebraic expression to describe the amount of rice remaining in Ramiro s pantry. Let x represent number of pounds of rice: x = x b. Determine if there are any constraints on x. x will be integer greater than zero but less than or equal (amount of rice available to use)

5 BEAT THE TEST! 1. José is going to have the exterior of his home painted. He will choose between Krystal Klean Painting and Elegance Home Painting. Krystal Klean Painting charges $ to come out and evaluate the house plus $7.00 for every additional 30 minutes of work. Elegance Home Painting charges $23.00 per hour of work. Let h represent the number of hours for which José hires a painter. Which of the following statements are true? Select all that apply. The expression 14h represents the total charge for Krystal Klean Painting. ý The expression 23h represents the total charge for Elegance Home Painting. The expression h + 23h represents the total amount José will spend for the painters to paint the exterior of his home. ý If José hires the painters for 10 hours, then Elegance Home Painting will be cheaper. ý If José hires the painters for 20 hours, then Krystal Klean Painting will be cheaper.

6 2. During harvesting season at Florida Blue Farms, hand pickers collect about 200 pounds of blueberries per day with a 95% pack-out rate. The blueberry harvester machine collects about 22,000 pounds of blueberries per day with a 90% pack-out rate. The pack-out rate is the percentage of collected blueberries that can be packaged to be sold, based on Florida Blue Farms' quality standards. Let h represent the number of days the hand pickers work and m represent the number of days the harvester machine is used. Which of the following algebraic expressions can be used to estimate the amount of collected blueberries that are packed at the Florida Blue Farms for fresh consumption this season? A 0.95h m B h (22000m) C h m D 200h m Answer is B. Want to learn more about how Florida Blue Farms uses algebra to harvest blueberries? Visit the Student Area in Algebra Nation to see how people use algebra in the real world! You can find the video in the Math in the Real World: Algebra at Work folder. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

7 3x 2 2y 4z 5 3 Section 1 Topic 2 Understanding Polynomial Expressions A term is a constant, variable, or multiplicative combination of the two. Consider 3x K + 2y 4z + 5. How many terms do you see? 4 List each term. This is an example of a polynomial expression. A polynomial can be one term or the sum of several terms. There are many different types of polynomials. A monarchy has one leader. How many terms do you think a monomial has? 1 A bicycle has two wheels. How many terms do you think a binomial has? 2 A triceratops has three horns. How many terms do you think a trinomial has?

8 Let s recap: Type of Polynomial Number of Terms Example Monomial 1 2x 5 Binomial 2 3x + 5 Trinomial 3 4a 2 + 2b + 3c Polynomial 1 or more 2m + 3n + 1 p Some important facts: Ø Ø The degree of a monomial is the sum of the exponents of the variables. The degree of a polynomial is the degree of the monomial term with the highest degree. Sometimes, you will be asked to write polynomials in standard form. Ø Ø Ø Write the monomial terms in descending degree order. The leading term of a polynomial is the term with the highest. degree The leading coefficient is the coefficient of the leading. degree

9 Let s Practice! 1. Are the following expressions polynomials? If so, name the type of polynomial and state the degree. If not, justify your reasoning. a. 8x K y S Yes, monomial, degree 5 b. KT U SV No, it is the quotient of variables c. S K xw 5x S + 9x X Yes, trinomial, degree 7 d. 10a Z b K + 17ab S c 5a X Yes, trinomial, degree 8 e. 2m + 3n^_ + 8m K n No, it has the quotient of a constant and variable.

10 Try It! 2. Are the following expressions polynomials? a. _ K a + 2bK l polynomial o not a polynomial b. 34 l polynomial o not a polynomial c. `a a U o polynomial l not a polynomial d. 2rs + s W l polynomial o not a polynomial e. xy K + 3x 4y^_ o polynomial l not a polynomial

11 3. Consider the polynomial 3x W 5x S + 9x X. a. Write the polynomial in standard form. 9x 7 + 3x 4 5x 3 b. What is the degree of the polynomial? 7 c. How many terms are in the polynomial? 3 d. What is the leading term? 9x 7 e. What is the leading coefficient? 9

12 BEAT THE TEST! 1. Match the polynomial in the left column with its descriptive feature in the right column. A. x S + 4x K 5x + 9 V I. Fifth degree polynomial B. 5a K b S I II. Constant term of 2 C. 3x W 9x S + 4x d VII III. Seventh degree polynomial D. 7a Z b K + 18ab S c 9a X VI IV. Leading coefficient of 3 E. x e 9x S + 2x X III V. Four terms F. 3x S + 7x K 11 IV VI. Eighth degree polynomial G. x K 2 II VII. Equivalent to 4x d + 3x W 9x S Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

13 Section 1 Topic 3 Algebraic Expressions Using the Distributive Property Recall the distributive property. Ø If a and b are real numbers, then a b + c = a b + a. c One way to visualize the distributive property is to use models. Consider (a + 3)(a + 2). 3! "!! " "! # #! $! " + &! + $ Now, use the distributive property to write an equivalent expression for (a + 3)(a + 2). a a + a a = a 2 + 2a + 3a + 6 = a 2 + 5a + 6

14 Let s Practice! 1. Write an equivalent expression for 3(x + 2y 7z) by modeling and then by using the distributive property. " #$ &'!!" ($ #)' 3 x + 2y 7z = 3 x + 3 2y 3 7z = 3x + 6y 21z 2. Write an equivalent expression for (x 3)(x 2) by modeling and then by using the distributive property. 3! #!! # #! $ $! %! # &! + % x 3 x 2 = x x x 2 3 x 3 2 = x 2 5x + 6

15 Try It! 3. Use the distributive property or modeling to write an equivalent expression for m + 5 m 3. 3! #!! $ #! % %! &%! $ + $! &% m + 5 m 3 = m m + m m = m 2 + 2m 15

16 BEAT THE TEST! 1. Students were asked to use the distributive property to write an equivalent expression for the expression x 5 x 2. Their work is shown below. Identify the student with the correct work. For the answers that are incorrect, explain where the students made mistakes. Student 1 Incorrect. Student did not correctly distribute. Student 2 Correct. Student 3 Incorrect. 5 2 = 10, not 10. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

17 Section 1 Topic 4 Algebraic Expressions Using the Commutative and Associative Properties What is 5 + 2? What is 2 + 5? 7 7 Does it matter which number comes first? No What is 9 2? What is 2 9? Does it matter which number comes first? No This is the commutative property. Ø Ø The order of the numbers can be changed without affecting the sum or. product If a and b are real numbers, then a + b = b + a and a b =. b a Does the commutative property hold true for division or subtraction? If so, give an example. If not, give a counterexample. No, and

18 Let s look at some other operations and how they affect numbers. Consider What happens if you put parentheses around any two adjacent numbers? How does it change the sum? (4 + 6) It does not change the sum. Consider What happens if you put parentheses around any two adjacent numbers? How does it change the product? (3 6) 4 3 (6 4) It does not change the product. This is the associative property. Ø Ø Ø The order of the numbers does not change. The grouping of the numbers can change and does not affect the sum or. product If a, b, and c are real numbers, then a + b + c = a + (b + c) and ab c =. a(bc) Does the associative property hold true for division or subtraction? If not, give a counterexample. No, and 20 (4 2) (20 4) 2.

19 Let s Practice! 1. Name the property (or properties) used to write the equivalent expression. a. [5 + 3 ] + 2 = 5 + [( 3) + 2] Associative property of addition b. (8 4) 6 = 6 (8 4) Commutative property of multiplication c. p + u + t = t + u + p Commutative property of addition When identifying properties, look closely at each piece of the problem. The changes can be very subtle.

20 Try It! 2. Identify the property (or properties) used to find the equivalent expression. a. (11 + 4) + 5 = (5 + 11) + 4 Commutative property of addition Associative property of addition b. v (y b) = (v y) b Associative property of multiplication c. (8 + 1) + 6 = 8 + (1 + 6) Associative property of addition d = Commutative property of multiplication 3. The following proof shows (3x)(2y) is equivalent to 6xy. Fill in each blank with either commutative property or associative property to indicate the property being used. 3x (2y) = 3 x 2 y = 3 2 x y = (3 2)(x y) = 6xy Associative Prop. Of Multiplication Commutative Prop. Of Multiplication Associative Prop. Of Multiplication

21 BEAT THE TEST! 1. Fordson High School hosted a math competition. A student completed the following proof and the question was marked incorrect. Identify and correct the student s mistake(s). Step 4 is incorrect. In Associative Property of Addition, the order of the numbers does not change. In step 4, the student changed the order of the terms in the sum. The correct answer is Commutative Property of Addition. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

22 Section 1 Topic 5 Properties of Exponents Let s review the properties of exponents. 2 W = = 16 2 S = = 8 2 K = 2 2 = 4 2 _ = 2 What pattern do you notice? We are dividing by 2 each time. Continuing the pattern, what does the following term equal? 2 q = = 2 2 = 1 Ø This is the zero exponent property: a q =. 1 Continuing the pattern, what do the following terms equal? 2^_ = = 1 2 = 1 2 = ^K = 2^1 2 = = = Ø This is the negative exponent property: a^r = a n and T _ a n st =. 1

23 Let s explore multiplying terms with exponents and the same base. 2 S 2 W = = e 2^S = x S x K = x x x x x = x = 2 2 = 22 Ø a m{n This is the product property: a u a r =. Let s explore dividing terms with exponents and the same base. W v = W w = 4 2 `x `y = x x x x x x x x x x x x x x x = 1 x = x^1 Ø This is the quotient property: Tz T t =. a m^n Let s explore raising powers to an exponent. (5 S ) K = = 5 3{3 = 5 6 y W S = y 4 y 4 y 4 = y 4{4{4 = y 12 Ø a mn This is the power of a power property: (a u ) r =.

24 Let s explore raising a product to an exponent. 2 3 K = = = x S = 4 x 4 x 4 x = x x x = 4 3 x 3 Ø a n b n This is the power of a product property: (ab) r =. Let s explore a quotient raised to an exponent. Kq S K 20 = = = Z a S 6 = 6 6 = 63 y y y y 3 Ø This is the power of a quotient property: a b a n n =. b n

25 Let s Practice! 1. Determine if the following equations are true or false. Justify your answers. a. 3 S 3 W = (39 ) (3 2 ) True = 3 3{4 = 3 7 and (39 ) (3 2 ) = 3(9^2) = 3 7 b. (5 4 K ) S = 5 W 5 q W e s} ^_ False (5 4 2 ) 3 = = ^1 = = =

26 Try It! 2. Use the properties of exponents to match each of the following expressions with its equivalent expression. A) X K W II I. 7 S 2 Z B) (7 2 K ) S I II. X ~ K ~ C) (7 K )(7 K ) IV III. K~ X ~ D) 7 K 7 q V IV. 7 W E) X K ^W III V. 7 K F) (X ) (X w ) VI VI. 7 S

27 BEAT THE TEST! 1. Crosby and Adam are working with exponents. Part A: Crosby claims that 3 S 3 K = 3 e. Adam argues that 3 S 3 K = 3 Z. Which one of them is correct? Use the properties of exponents to justify your answer. Crosby is correct = 3 3{2 = 3 5. Adam multiplied the exponents instead of adding. Part B: Crosby claims that Sy S y S U = 34. Adam argues that S U = 36. Which one of them is correct? Use the properties of exponents to justify your answer. Adam is correct. 38 = 38^2 = 3 6. Crosby divided the 3 2 exponents instead of subtracting. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

28 Section 1 Topic 6 Radical Expressions and Expressions with Rational Exponents Exponents are not always in the form of integers. Sometimes you will see them expressed as rational numbers. Consider the following expressions with rational exponents. Use the property of exponents to rewrite them as radical expressions. 9 } U = = = 3 8 } w = = = 2 Do you notice a pattern? If so, what pattern did you notice? = 3 = 9 and = 2 = 8 Consider the following expression with rational exponents. Use the pattern above and the property of exponents to rewrite them as radical expressions. 2 U 1 w = = 2 2 or = w U = 5 3 or = = 5 3 For exponents that are rational numbers, such as T V, we b a a b a ƒ a have x = =.

29 Let s Practice! 1. Use the rational exponent property to write an equivalent expression for each of the following radical expressions. a. x + 2 x w b. x x Use the rational exponent property to write each of the following expressions as integers. a. 9 } U 9 = 3 b. 16 } U 16 = 4 c. 8 } w d. 8 U w 3 8 = = 2 2 = 4 e. 125 U w = 5 2 = 25 f. 16 w ~ = 2 3 = 8

30 Try It! 3. Use the rational exponent property to write an equivalent expression for each of the following radical expressions. a. y y 1 2 v b. y y Use the rational exponent property to write each of the following expressions as integers. a. 49 } U 49 = 7 b. 27 } w 3 27 = 3 c. 216 U w = 6 2 = 36

31 BEAT THE TEST! 1. Match each of the following to its equivalent expression. A. 2 } w VI 1. m } U 3 B. m 3 III 2. (3m) } U C. 2 U w V 3. (m 3) } U D. m 3 I 4. 2 E. 2 } w U IV 5. 4 w F. 3m II 6. 2 Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

32 Section 1 Topic 7 Adding Expressions with Radicals and Rational Exponents Let s explore operations with radical expressions and expressions with rational exponents. For each expression, label approximately where the answer would be found on the number line _ K + 2 _ K Since the radicands are not the same, we cannot add the radicals. Since the bases are not the same, we cannot add exponents } U + 5 } U } U 8 3 } U To add radicals, the radicand of both radicals must be the same. To add expressions with rational exponents, the base and the exponent must be the same. In both cases, you add only the coefficients.

33 Let s Practice! 1. Perform the following operations. a = = = 3 3 b. 12 } U + 3 } U = = = = c = = = d. 72 } U + 15 } U + 18 } U = = = = w e = = f. 32 } U + 16 } w = = = For radicals and expressions with rational exponents, always look for factors that are perfect squares when taking the square root (or perfect cubes when taking the cube root).

34 Try It! 2. Perform the following operations. a = 4 6 c = = = b. 6 } U } U = d. 50 } U + 18 } U + 10 } U = = = = e. w 2 = 3 2 = = 3 2 w w f. 2 } w + 8 } w + 16 } w = = = =

35 BEAT THE TEST! 1. Which of the following expressions are equivalent to 3 2? Select all that apply. o 3 } U + 2 } U ý 8 } U + 2 } U ý 3 2 } U ý 18 o 2 18 ý Miguel completed a proof to show that = 4 3 } U : = 27 } U + 3 } U = = 3 3 } U + 3 } U = 4 3 _ K Part A: Which expression can be placed in the blank to correctly complete Miguel s proof? A 3 } U 9 } U + 3 } U B C 9 } U + 3 } U + 3 } U D 9 } U + 3 } U Answer: B

36 Part B: Label and place 4 3 } U on the number line below _ K = Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

37 Section 1 Topic 8 More Operations with Radicals and Rational Exponents Let s explore multiplying and dividing expressions with radicals and rational exponents = 10 2 = 20 = 4 5 = _ K 2 _ K = = ( ) = ( ) = { = w 2 2 We cannot multiply the radicals since the roots are not the same. 2 _ K 2 _ S = { 1 3 = = 10 2 = 5 10 _ K 2 _ K = = =

38 The properties of exponents also apply to expressions with rational exponents. Let s Practice! 1. Use the properties of exponents to perform the following operations. a. x } w } U = x = x 6 b. 7 S = 7 3 = = 7 7 c. a } Ub U v a U wb } U = a 1 2 a 2 3 b 2 5 b 1 2 = a 1 2 { 2 3 b 2 5 { 1 2 = a 7 6 b 9 10 = a 1{1 6 b 9 10 = a 6 a 10 b 9

39 d. ~ = = = = = Try It! 2. Use the properties of exponents to perform the following operations. a. (m q n K ) } v = m n = m 0 n 2 5 = 1 n 2 5 = n 2 5 w b. 8 3 U w = = = = = = 2 9

40 ~ w c. 4 4 = = { 1 3 = = d } U = = = = = 4 3 3

41 BEAT THE TEST! 1. Which of the following expressions are equivalent to 2 } U? Select all that apply. g. w 4 h. w 8 ý ~ 4 ý 8 i. 16 w w ~ 4 = 4 _ S = (2 K ) _ S = 2 K S 8 = 8 _ S = (2 S ) _ S = 2 4 = 4 _ W = (2 K ) _ W = 2 _ K 8 = 8 _ Z = (2 S ) _ Z = 2 _ K 16 = 16 _ Z = (2 W ) _ Z = 2 K S Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

42 Section 1 Topic 9 Operations with Rational and Irrational Numbers Let s review rational and irrational numbers. Ø Ø Numbers that can be represented as T,where a and b V are integers and b 0, are called rational numbers. Numbers that cannot be represented in this form are called irrational numbers. o Radicals that cannot be rewritten as integers are examples of such numbers. Determine whether the following numbers are rational or irrational. Rational Irrational 9 l 8 l π l 22 7 l l 33 2 l l 25 l

43 Given two rational numbers, a and b, prove that the sum of a and b is rational. The sum is always rational. Examples: = 8, = 5 6 Given two rational numbers, a and b, what can be said about the product of a and b? The product is always rational. Examples: 3 5 = 15, = 0. 51

44 Given a rational number, a, and an irrational number, b, prove that the sum of a and b is irrational. The sum is always irrational. Examples: = , 2 + 3π = 2 + 3π Given a non-zero rational number, a, and an irrational number, b, what can be said about the product of a and b? The product is always irrational. Examples: 2 5, 4π

45 Let s Practice! 1. Consider the following expression The above expression represents the l sum o product of a(n) l rational number o irrational number and a(n) o rational number l irrational number and is equivalent to a(n) o rational number l irrational number

46 Try It! 2. María and her 6 best friends are applying to colleges. They find that Bard College accepts _ of its applicants. María S and her friends write the expression below to represent how many of them would likely be accepted The above expression represents the o sum lproduct of a(n) lrational number o irrational number and a(n) lrational number o irrational number and is equivalent to a(n) lrational number. o irrational number.

47 BEAT THE TEST! 1. Let a and b be non-zero rational numbers and c and d be irrational numbers. Consider the operations below and choose whether the result can be rational, irrational, or both. Rational Irrational a + b ý a c ý a b ý c d ý ý c + d ý ý a b c ý 2. Consider x y = z. If z is an irrational number, what can be said about x and y? At least one of the variables MUST BE irrational. Test Yourself! Practice Tool Great job! You have reached the end of this section. Now it s time to try the Test Yourself! Practice Tool, where you can practice all the skills and concepts you learned in this section. Log in to Algebra Nation and try out the Test Yourself! Practice Tool so you can see how well you know these topics!

48 THIS PAGE WAS INTENTIONALLY LEFT BLANK