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Math 7 2016-2017 Packet #3 Expressions and Equations (Module 3) Ms. Pricola, Mr. Unson 1 Table of Contents: Review of Integer Rules Vocabulary of Numbers – Natural/Whole/Integers/Rational Review of Properties Vocabulary (Expression / Equation / Constant/Variable/ Coefficient/Term/Like Terms/Monomial/Polynomial/ Binomial/Trinomial) Combining Like Terms Vocabulary (Factor / Greatest Common Factor / Factoring) Factoring out the Greatest Common Factors (GCF) More Combining Like Terms Writing Expressions Writing Equations Adding and Subtracting Expressions Multiple Step Word Problems One Step Equation Solving (and Checks) Two Step Equation Solving (and Checks) Solving Equations by Combining Like Terms on Same Side Solving Equations using the Distributive Property Multiple Step Equation Solving Multiple Step Word Problems Representing Inequalities on number line One Step Inequality Solving Two Step Inequality Solving Word Problems using Inequalities Mixed Review of the entire Domain - Expressions and Equations Review for Quizzes and Tests Review of all Domains covered so far Ratios and Proportional Relationships The Number System Expressions and Equations Enrichment Section (*Intro now, continue after State Assessment ** After State Assessment) Topics will include: Solving Inequalities using the negative rule *Rules of Exponents *Square and Cube Roots *Scientific Notation *Solving Equations with Variables on Both Sides **Solving Simultaneous Equations Using Graphing **Solving Simultaneous Equations Algebraically Page 3 Page 4 Pages 5 - 7 Page 8 Pages 10 - 11 Page 12 Pages 13 - 15 Pages 16 - 18 Pages 19 - 22 Pages 23 - 24 Pages 25 - 27 Pages 28 - 30 Page 31 Pages 32 - 33 Page 34 Page 35 Page 36 - 39 Page 40 - 42 Pages 43 - 44 Page 45 Page 46 Pages 47 - 50 Pages 51 - 58 Pages 59 - 69 Pages 70 – 74 Pages 75 – 85 Pages 86 - 98 Pages 99 –108 2 In this packet: * Questions with this annotation were taken from the North Carolina CCSS ** Questions with this annotation were taken from the Utah CCSS Integer Rules Additon 1. Same Sign – Add and keep the sign of the numbers Ex. -6 + (-4) = -10 2. Different Signs – Subtract and keep the sign of the number with the greater absolute value Ex. -8 + 2 = -6 -4 + 9 = 5 Subtraction You have to change the problem!!!! You must change it to “add the opposite” then use the two rules for addition Ex. -8 – 5 change to -8 + -5 and your answer is -13 -5 – (-2) change to -5 + 2 and your answer is -3 6 – (-4) change to 6 + 4 and your answer is 10 Multiplication and Division – are the same two rules instead of a multiplication sign you can have a division sign Rule #1 negative x negative = positive Rule #2 negative x positive = negative Examples: -8 x -2 = 16 (n x n = p) (n x p = n) -4 x 3 = -12 3 -9 ÷ -3 = 3 -8 ÷2 = -4 Natural Numbers – the set of Counting Numbers: 1, 2, 3, 4, 5,…….. Whole Numbers – the set of Natural numbers and zero: 0, 1, 2, 3, 4, 5, ……. Intergers – Positive whole numbers, negative whole numbers, and zero: ……–2, –1, 0, 1, 2, ……. Rational Numbers – Numbers that can be written as fractions. For example, –5, 0, 12, , 0.233333… 4 Review of Properties: Commutative Property – the order in which numbers are added has no effect on their sum. This is also true for multiplication; numbers may be multiplied in any order and the resulting product will be the same. Examples: 2+3+5= 3+5+2 4 x 5 =5 x 4 a+b=b+a Associative Property – the way in which numbers are grouped will have no effect on their sum. This is also true for multiplication; numbers may be grouped in any order and the resulting product will be the same. The order of the values does not change. Examples: (2 + 3) + 5 = 2 + (3 + 5) ( ) 4 x (5 x 2) = (4 x 5) x 2 ( ) (a + b) + c = a + (b + c) Distributive Property – When multiplying a number by a sum, you can multiply the number by the first value in parenthesis, then add that product to the product of the number by the second value in parenthesis. The resulting sum will be the same as the sum generated by adding the values in the parenthesis first, and then multiplying by the number. Examples: 4(2 + 3) = 4(2) + 4(3) ( ) Zero Property of Multiplication – any number times zero has a product of zero. Also, zero times any number has a product of zero. Examples: 3x0=0 0 x 14 = 0 Identity Property of Addition (Additive Identity)– when zero is added to any value, the resulting sum is that original value. Also, adding a value to zero will result in a sum of that value. Examples: 85 + 0 = 85 0 + 93 = 93 Identity Property of Multiplication (Multiplicative Identity)– when one is multiplied by any number, the resulting product is that number. Also, when multiplying a number by one, the resulting product is that original number. Examples: 24 x 1 = 24 1 x 47 = 47 Additive Inverse Property – When a number and its inverse are added, the sum is zero. 5 Examples: 5 + -5 = 0 -125 + 125 = 0 Multiplicative Inverse Property – Two numbers are said to be multiplicative inverses when they have a product of 1. ( ) Examples: Practice Using Properties Name the property being used to make each line change: 1] ( [2] ___________________ ( 3] ) ___________________ ) [4] ( ) __________________ ( ) ___________________ 12n – 12n + 36 ___________________ 36 ____________________________ _____________________________ _____________________________ 5] ( ) _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ 6 6] ( ) ( ) ( ________________________ ) ________________________ ________________________ ________________________ 7] Rewrite using the distributive property. Do not simplify the resulting expression. = _____________________________ 8] Rewrite the following using the distributive property and then find the sums. -4.8x + (-6.2x) 9] What is the missing addend in the following: 2.34y + _________ = 0 7 Vocabulary Expression – a series of one or more terms, not containing an equal sign. Examples: 4n + 3f 8 + 7g -5y – 10 + 6k Equation – a series of terms containing an equal sign. Examples: 2a = 14 -50n + 10 = -40 p+8=9 Constant – a mathematical value which never changes. Examples: 3 , -5 , , π Variable – a mathematical “placeholder” which may be assigned any value. Once a value is assigned, it must remain that value for the entire problem. Examples: Coefficient – a constant placed before (and adjacent) a variable. The coefficient is being multiplied with the variable. In the following example, the “3” is the coefficient: Example: 3f 3 is the coefficient Term – consists of either a constant, variable, or combination of both being multiplied. Terms are separated by addition or subtraction signs. Example: If 4 + n + 5g is the expression then 4 , n , 5g are the terms 8 Like Terms – Those terms having the same variable and exponent. Note: the coefficients do not have to be the same, although they may be. Examples: 3n & 5n -4f & 120f Monomial – An expression with only one term. That term can be a real number, a variable, or a product of a real number and variables with whole number exponents. Examples: 3, x, 5xy, Polynomial – is a monomial or the sum or difference of monomials Binomial – A polynomial with 2 terms. Examples: x – 3, 5x + 1, Trinomial – A polynomial with 3 terms. Example: 9 In each of the following, name the; terms, constants, coefficients, and like terms. Terms Constants Coefficients Like Terms 7x + 4x + 3y + 2 3x + 4 + 2x + 3 3m – 2n + 5m – 4 6 + 2x + 4x -10x 9m + 2r – 2m + r In each of the following, combine the like terms: 1] 8x + 5x = _____________ [2] 10y – 4y = ______________ 4] 6y + 4x + 2y + 3x = ________________ 6] 8x + 8y + 3x = ____________ [5] 3x + 2 + 4x + 5 = ____________________ [7] 4x – 3 + 5x = ____________ 9] x – 3x + 2x + 4 = ____________________ 11] 5x + 8 – 7x = ___________ [8] -5x + 8x = ___________ [10] 4x + 3 – 5x + x = _____________________ [12] -6x – 8x = ____________ 14] 6x + 4 – 2x – 3 = ________________ 16] -7y – 12y = __________ [3] 4 + 3x + 2 = __________ [13] 4x – x = __________ [15] 7y + 5x – 9y + 6x = ___________________ [17] 4x + 3 – 10 – 2x = ______________________ 18] 2x + x + 3x – 4x = ___________________ [19] 12 – 4x + 7y – 9x – y = __________________ 10 20] 6x + 12 – 30x – (-8) = _________________ [21] 8 – (-2) + 5n – (-15n) = __________________ 22] 4x – 7x + 2(x + 6) + (–4) = _____________________________________ 23] –5x + 2x + 3(2x + 4y) + y = _____________________________________ 24] 2(x – 3y) + 4(-3x + 2y) = _________________________________________ 25] (3x + ) – 5x + 4 = ______________________________________ 11 Vocabulary Factor – one integer is a factor of another integer if it divides that integer with a remainder of zero. Example: 2 is a factor of 10 Prime Number – positive integer greater than one with exactly two factors 1 and itself. Example: 13, 17, 2, 5 Composite Number – positive integer greater than one with more than two factors. Example: 22, 36, 100 Greatest Common Factor (GCF) – the largest number that is a common factor of 2 or more numbers or expressions. Example: Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 16: 1, 2, 4, 8, 16 GCF is 4 Factoring – Writing a polynomial as the product of 2 factors by using the GCF of the terms as one of the factors. Factoring is the reversing the Distributive Property. Example: Factor 6xy + 12x Answer: 6x(y + 2) 12 Using the Greatest Common Factor (GCF) Find the GCF of the following: A] 3a + 4a ; gcf = _________ B] 12b + 4c ; gcf = ________________ C] 10 – 2c + 4d ; gcf = _____________ D] d + 4d + 5d ; gcf = ________________ E] 6f + 12c – 3 + 18d ; gcf = _________ F] 4w – 6w + 10w ; gcf = __________ Rewrite each of the following using the GCF as a product of two factors: 1. 15a + 5 = _________________ 2. 16d – 6 = ________________ 3. 20k + 14m – 2 = _______________ 4. 18h – 8f + 4 = ___________________ 5. 20 – 10y = _________________ 6. 18 + 9c + 27a = ____________________ 7. 4w + 12k – 10p = _______________ 8. 17r + 51a = __________________ 9. -7p + 14g = ________________ 10. -7p – 14g = _________________ 13 11. -6w + 10r – 4 = _______________ 12. -6w – 10r + 4 = ____________________ 13. Jerry has 4 jelly beans, Suzi has 12, and Billy 8. What is the largest number of jelly beans they could each use to make equal numbered piles? (All 3 kids must have the same number in each pile) Factoring Out GCF 1. (Taken from NYS Testing Draft 7.EE question #2) Which expression below is equivalent to + ? A ( ) B ( + ) C ( + ) D ( + ) + 2.* Write equivalent expressions for: 3a + 12 3.* An equilateral triangle has a perimeter of 6x + 15. What is the length of each side of the triangle? 4.** Factor: −3x + 9 5.** Which students correctly simplified the expressions? Justify your reasoning. Fix all incorrectly simplified expressions. Brianda: = Sara: = Jorge: = 14 Julia: = Trent: = 6. Factor the following completely: 12x + 8xy 7. The delivery person for a flower shop earns a salary of $40 a day plus $3 for each delivery he makes. As an expression he would earn (40 + 3d) a day. Write an expression that would show how much the delivery person will make if he works seven days delivering flowers. Expand your expression and explain your work. Another flower shop offers the delivery person another deal. They will pay him (90 + 6d) dollars total for 3 days work. Will this job pay more or less than his current job. Use factoring to compare the two offers. 15 More Combining Like Terms 1.* Suzanne says the two expressions 2(3a – 2) + 4a and 10a – 2 are equivalent. Is she correct? Explain why or why not. 2. While doing your homework your friend suggests that -6(2x + 9) can be written as -12x + 54. Do you agree with your friend or suggest a different answer? Explain. 3. In a given Isosceles triangle, the base is found to be half the length of one of the sides. The expression for the perimeter is given as . Write the perimeter in simplest form. 4.** Simplify the following linear expression: + ( − 7) 5. Find the sum + 6. Explain why the sum of 6x and 8y is not equal to 14xy 7. Simplify the following expressions. Use number properties and rules to explain each step. 16 A − + [− + ( ) B 6x + 4(2 – x) + 3(2x – 1) 8. Can 2.3x + 4.5y + 1.4 be simplified further? Explain why or why not 9. Simplify the following and explain what properties or rules allow you to do this + + +( ) 10. Why must you use the distributive property as your first step in simplifying the following expression? 2y + 4(3y – 2) 11. Why must you simplify the following expression first before you can factor it? Explain why then factor it. 2x + 8y + x + 7y 17 12. Find the perimeter of the following triangle 2x + 9 5x − 12 x + 20 13. Simplify 14. Simplify 15. Simplify 16. Is 5x + 8 – (4 + 7x) ( ) (not the same as #13) 6 – 7(2d – 5) -35 90 + 3(40h – 60) + (-100h) = 30(4h + 3) ? 18 17. Joel has two “x” magnets for his refrigerator and $3. Sarah has three times as much as Joel of both “x” magnets and money. How much do they have together? (x + 3) + 3(x + 3) = Writing Equivalent Expressions Write the following in two different ways: 1] w + w + 2w + 2w 2] 8p + 2p + 8p + 2p __________________________________ __________________________________ __________________________________ __________________________________ 3] A rectangle is three times as long as it is wide. One way to write an expression to find the perimeter would be w + w + 3w + 3w. Write the expression in two other ways. __________________________________ __________________________________ 4] Write an equivalent expression for 3(x + 5) – 2. Explain your steps. 5] Ralph reached into his pocket and found some change. He yelled out “Hey! I just found a quarter, and two dimes, and another quarter, and three more dimes!”. Write this as an algebraic expression in two different ways. __________________________________ ____________________________________ 19 6] Write an equivalent expression for 5(n – 4) – 30. _________________________ 7] Which is not an equivalent expression to 7b – 3(b – 4) ? a) 7b – 3b + 12 b) 4b + 12 c) 4b – 12 d) 4(b + 3) 8] A balloon started from sea level, rose 5 meters, dropped 2 meters, dropped another meter, then rose 8 meters. Write this as a variable expression, then write an equivalent expression. ___________________________________ ____________________________ Writing Expressions Write an expression for each situation: 1. Four less than a number “x”. ____________________________ 2. Seven more than a value “n”. ____________________________ 3. Five times p pieces of paper. ____________________________ 4. D dollars divided 6 ways. ____________________________ 5. Twice a value r plus ten. ____________________________ 6. Eight less than twice a value g. _____________________________ 7. Nine, less a number k. _____________________________ 8. Fourteen divided by a number s. _____________________________ 9. Three more than four times n. _____________________________ 10. Five times the sum of y and 6. _____________________________ 11. A flower shop owner makes $2 on each flower sold. The shop pays $280 in rent each day for the store. After the cost of renting the store is deducted, the expression that shows what the shop owner earns would be (2f – 280) per day. Which expression shows how many dollars the owner earns in a 7 day week? 20 A B C D 266 14f – 280 14f – 1960 1946f 12. The delivery person for a flower shop earns a salary of $40 a day plus $3 for each delivery he makes. The expression that shows what he would earn in a day would be (40 + 3d) . Write an expression that would show how much the delivery person will make if he works seven days delivering flowers. Write an expression for 10 days: _____________ Write an expression for 14 days: _____________ 13.* All varieties of a certain brand of cookies are $3.50. A person buys peanut butter cookies and chocolate chip cookies. Write an expression that represents the total cost, T, of the cookies if p represents the number of peanut butter cookies and c represents the number of chocolate chip cookies. What would this expression look like if the cookies cost $3.75? ________________ 14.* Jamie and Ted both get paid an equal hourly wage of $9 per hour. This week, Ted made an additional $27 dollars in overtime. Write an expression that represents the weekly wages of both if J = the number of hours that Jamie worked this week and T = the number of hours Ted worked this week. What is another way to write this expression? Write an expression if no one made any additional money from overtime: ___________ 15.* Given a square pool as shown in the picture, write four different expressions to find the total number of tiles in the border. Explain how each of the expressions relates to the diagram and demonstrate that the expressions are equivalent. Which expression is most useful? Explain. 21 16. Which of the following expressions has the same meaning as “increase y by 30%” A B C D 0.30y 0.70y 1.30y 1.70y 17. What does the expression 0.45x represent? A B C D increase x by 55% decrease x by 55% increase x by 45% decrease x by 45% 18.** The students in Mr. Sanchez's class are converting distances measured in miles to kilometers. To estimate the number of kilometers, Abby takes the number of miles, doubles it, then subtracts 20% of the result. Renato first divides the number of miles by 5, then multiplies the result by 8. 1. Write an algebraic expression for each method. 2. Use your answer to part (a) to decide if the two methods give the same answer. 22 19. (Taken from www.illustrativemathematics.org) Write an expression for the sequence of operations. 1. Add 3 to x , subtract the result from 1, then double what you have. 2. Add 3 to x , double what you have, then subtract 1 from the result. 20. The Smith family sold their house for d dollars. The real estate agent gets a 2% commission on the sale price. Write an expression, in simplest form, to represent how many dollars the Smith family will receive from the sale of their house after the agent’s commission is deducted from the sale price. Explain your work. Writing Equations 1.* The youth group is going on a trip to the state fair. The trip costs $52. Included in that price is $11 for a concert ticket and the cost of 2 passes, one for the rides and one for the game booths. Each of the passes cost the same price. Write an equation representing the cost of the trip and determine the price of one pass. How much would each pass cost if the total $52 was the same, except there were 4 passes? Write an equation to determine this situation. 2.* Amy had $26 dollars to spend on school supplies. After buying 10 pens, she had $14.30 left. How much did each pen cost including tax? Write a sentence to find the cost of one pen. ___________________________ 23 What would the equation be if the total cost were $40 and Amy bought 24 pens? __________________ 3.* The sum of three consecutive even numbers is 48. What is the smallest of these numbers? 4.** John and his friend have $20 to go to the movies. Tickets are $6.50 each. How much will they have left for candy? Solve this question in two ways. 5. Jay’s son is not very good at handling money. He likes to spend and he doesn’t like to save. His bank debit card had a balance of $423.52. After three purchases, each for the same amount, his balance was −$77.32. What was the amount , p, of each purchase he made. Write an equation and solve it to find the answer. 6. Sam took a taxi while visiting his daughter in Boston and was charged an initial fee of $4.50 plus $1.75 for every mile I traveled. The ride ended up costing me $53.50. Write an equation and solve it to find how many miles, m, Sam travelled in the taxi. 24 7. The perimeter of a rectangle is 144 inches. The length of the rectangle is 32 inches. Write an equation and solve it to find the width, w, of the rectangle. 8. During a recent cookie sale, the cost of a box of cookies was $4.50. The organization gets to keep 20% of that cost, but they then donate $0.20 per box to charity. Write an equation to find the number of boxes sold if their final profit was $160. Adding and Subtracting Algebraic Expressions For each of the following perform the indicated operation: 1. (3x + 6y) + (2x + 8y) = _______________________________ 2. (5x – 4) + (–2x + 6) = ________________________________ 3. (–8x – 4y) + (–2x + 3y) = _____________________________ 4. (4x – 3) + (–6x – 7) = _________________________________ 5. ( ) ( ) = ___________________________ 25 6. (3x + 6y) – (2x + 8y) = _______________________________ 7. (5x – 4) – (–2x + 6) = ________________________________ 8. (–8x – 4y) – (–2x + 3y) = _____________________________ 9. (4x – 3) – (–6x – 7) = _________________________________ ( 10. ) – ( ) = ___________________________ Adding / Subtracting Algebraic Expressions 1. (Taken from NYS Testing Draft 7.EE question #1) When + is subtracted from − A − B − C + D + 2. What is the difference when , the result is is subtracted from ? 26 3. What is the difference when is subtracted by 4. What is the sum of and ? ? Find the sum of each the following: ( ) ( ( ( ) ) ) ( ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) 27 Find the difference of each of the following: ( ( ( ) ( ) ( ) ( ) ) ) ( ) ( ) ( ) ( ) ( ) ( ) Multiple Step Word Problems 1. John buys a shirt for “s” dollars. He must pay 6% sales tax on his purchase. Which expression represents how many dollars he will pay in all, including sales tax? A B C D 1.6s 1.06s 6s s + 0.06 2.* Three students conduct the same survey about the number of hours people sleep at night. The results of the number of people who sleep 8 hours a night are shown below. In which person’s survey did the most people sleep 8 hours? Susan reported that 18 of the 48 people she surveyed get 8 hours sleep a night Kenneth reported that 36% of the people he surveyed get 8 hours sleep a night Jamal reported that 0.365 of the people he surveyed get 8 hours of sleep a night 28 3.** Malie and her sister won a $45 iTunes gift card. They agree to split the money so that Malie gets of the value and her sister gets the rest. If songs on iTunes cost $1.29, how many songs will each sister be able to buy? 4.** Braxton wants to spend his $60 savings on new longboard parts online. He has a promotional code that he can use for off his cost before shipping or for free shipping. If shipping costs are $1.75 for each $10 spent, how should he use his promotional code? Justify your answer. (Taken from www.illustrativemathematics.org) 5. Katie and Margarita have $20.00 each to spend at Students' Choice book store, where all students receive a 20% discount. They both want to purchase a copy of the same book which normally sells for $22.50 plus 10% sales tax. To check if she has enough to purchase the book, Katie takes 20% of $22.50 and subtracts that amount from the normal price. She takes 10% of the discounted selling price and adds it back to find the purchase amount. Margarita takes 80% of the normal purchase price and then computes 110% of the reduced price. Which student is correct? Do they have enough money to purchase the book? 29 6. When working on a report for class, Catrina read that a woman over the age of 40 can lose approximately 0.16 centimeters of height per year. 1. Catrina's aunt Nancy is 40 years old and is 5 feet 7 inches tall. Assuming her height decreases at this rate after the age of 40, about how tall will she be at age 65? (Remember that 1 inch = 2.54 centimeters.) 2. Catrina's 90-year-old grandmother is 5 feet 1 inch tall. Assuming her grandmother's height has also decreased at this rate, about how tall was she at age 40? Explain your reasoning. 7. In Sue’s son’s baby book Sue recorded that he was recorded that he grew an average of inches tall on his 2nd birthday. She also inches each year for the next 12 years. Then he grew an average of 1 inches each year until his 18th birthday. According to Sue’s record keeping how tall would her son have been when he was blowing out those birthday candles at age 18? Is this reasonable? Explain why or why not. 30 8. Students in technology class had the opportunity to create a family name sign. Each student chose either a ⁄ foot board, a ⁄ foot board, or a ⁄ foot board. If nine students chose the ⁄ , seven chose the to have? ⁄ , and six the ⁄ foot long piece, how much wood did the tech teacher need One Step Equation Solving A] B] C] D] E] F] 31 G] J] H] ( ) K] I] ( ) L] M] N] O] P] -100p = -1 Q] R] Two Step Equation Solving 1. * − 4 = −16 3. −4y + 9 = −3 2. ( ) 4. 32 5. 6. 7. 8. 9. 10. 33 11. 12. 13. 14. 15. 16. ( ) Solving Equations by Combing Variables on the Same Side 1. 2. 34 3. 4. 5. 6. 7. 8. ( ) Solving Equations Using the Distributive Property 1. ( ) 2. ( ) 35 3. ( ) 4. 5. ( ) 6. 7. ( ) 8. ( )) ( ( ) ) Multiple Step Equation Solving Solve each of the following showing the proper algebra steps: 1. 3(x – 4) + 15 = 21 2. 2(5x + 4) – 6x = 28 36 3. 8x + 2(x – 3) + 4 = ─22 4. 5. (x + 12) + 7x ─ 3(2x ─ 4) = ─9 x ─ 2 = ─6 More practice Equation Solving and Checks: Directions: Answer each question showing each step and a check: 1. –4x = 2.4 Check: 2. 5x – 7 = –37 Check: 37 3. Check: 4. 8p – 10p = – Check: 5. 12c + 3c – 14 = 31 Check: 6. Check: 7. 12c + 3c = 60 Check: 8. 9. ( ) Check: 9(x + 2) = 47.7 Check: 10. –9(x + 2) = 27 Check: 38 11. 4(x + 2) + 10 = –2 Check: 12. 4x – 9(x + 2) = 2 Check: 13. –2 + 4(3x + 5) = 5 Check: 14. Write an algebraic example of the Distributive Property 15. Jane is selling 3 paintings at (x + 14) dollars each. If the total of the 3 paintings comes to $60, find the value of “x”. Use an algebraic equation. 16. A TV is hung on an 8 foot high wall such that the center of the TV rests 5 feet above the floor. If the TV has a frame which is 30 inches high, how far from the ceiling would the top of the TV frame be? 39 Multiple Step Word Problems 1. (Taken from NYS Testing Draft 7.EE question #3) At a discount furniture store, Chris offered 40 a salesperson $600 for a couch and a chair. The offer includes the 8% sales tax. If the salesperson accepts the offer, what would be the price of the furniture, to the nearest dollar, before tax? A B C D $552 $556 $592 $648 2. (Taken from NYS Testing Draft 7.EE question #4) A framed picture 24 inches wide and 28 inches high is shown in the diagram below. The picture will be hung on a wall where the distance from the floor to ceiling is 8 feet. The center of the picture must be feet from the floor. Determine the distance from the ceiling to the top of the picture frame. Show your work. 3. Explain why a 20% discount is the same as finding 80% of the cost, c (0.80c) 4.* Amy had $26 dollars to spend on school supplies. After buying 10 pens, she had $14.30 left. How much did each pen cost including tax? 41 5.* The sum of three consecutive even numbers is 48. What is the smallest of these numbers? 6.** John and his friend have $20 to go to the movies. Tickets are $6.50 each. How much will they have left for candy? 7. Ken is not very good at handling money. He likes to spend and he doesn’t like to save. His bank debit card had a balance of $423.52. After three purchases, each for the same amount, his balance was −$77.32. What was the amount , p, of each purchase he made. Write an equation and solve it to find the answer. 8. Sue took a taxi while visiting her daughter in Boston and was charged an initial fee of $4.50 plus $1.75 for every mile she traveled. The ride ended up costing her $53.50. Write an equation and solve it to find how many miles, m, Sue travelled in the taxi. 9. The perimeter of a rectangle is 144 inches. The length of the rectangle is 32 inches. Write an equation and solve it to find the width, w, of the rectangle. 42 43 Representing Inequalities on the Number Line 44 Write an inequality to represent each graph 45 Solving One Step Inequalities Solve and Graph 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 14. 15. 13. ( ) ( ) 46 Solving Two Step Inequalities – Solve and graph 1. 3. 2. ( ) 4. 5. 6. 7. 8. 47 Word Problems with Inequalities A] Jerry needs to buy his sister a birthday gift, and he also needs to buy a few bottles of soda which cost $1.28 each. The birthday gift he has in mind will cost him $43.52. What is the greatest number of soda bottles Jerry can buy if he has $60? B] Samantha has $56. What is the maximum number of sunglasses she can buy if they cost $9.45 each? C] D] In a video game, every banana is worth 80 points. Cara wants to break the high score of 760 points. What is the least number of bananas she will need to peel? A wooden plaque template calls for inches of material (including the cut). If Jar Jar has a 3 foot board, how many plaques can he make? 48 1. Mrs. Pricola’s cell phone company charges her $48.80 a month, plus $0.03 for each text message she sends. Mrs. Pricola likes to text but she is trying to keep her monthly bill under $75. Write an inequality an solve it to find the number of text messages, t, that Mrs. Pricola can make and stay within her budget. 2. Jennifer went to the store with $40 to buy hamburger meat and rolls for a barbecue. The rolls cost $6. If the hamburger meat is $2.79 a pound, write and solve an inequality to show how many pounds of hamburger meat Jennifer can buy. 3. Chris earns $45 a day plus $6 for every sale he makes at the store he works in. He needs to earn at least $125 today to pay back a loan. Write an inequality and solve it to find out how many sales, s, Chris needs to make. 4. Jonah gives $50 to his favorite charity plus $0.30 a day to a local charity. Jonah would like to donate at 49 least $75 this years. How many days will he need to give $0.30 to achieve this goal? Write an inequality and solve it to find out how many days, d, he needs to donate. 5. (Taken from www.illustrativemathematics.org) Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry 1200 pounds of people and gear for safety reasons. Assume the average weight of a person is 150 pounds. Each group will require 200 lbs of gear for the boat plus 10 lbs of gear for each person. A. Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set. B. Several groups of people wish to rent a boat. Group 1 has 4 people. Group 2 has 5 people. Group 3 has 8 people. Which of the groups, if any, can safely rent a boat? What is the maximum number of people that may rent a boat? 50 6. (Taken from NYS Testing Draft 7.EE question #5) Mandy’s monthly earnings consist of a fixed salary of $2800 and an 18% commission on all her monthly sales. To cover her planned expenses, Mandy needs to earn an income of at least $6400 this month. Part A: Write an inequality that, when solved, will give the amount of sales Mandy needs to cover her planned expenses. Answer: ___________________ Part B: Graph the solution of the inequality on the number line. 7. (Taken from NYS Testing Draft question #9) When John bought his new computer, he purchased an online computer help service. The help service has a yearly fee of $25.50 and a $10.50 charge for each help session a person uses. If John can only spend $170 for the computer help this year, what is the maximum number of help sessions he can use this year? 51 8.* Florencia has at most $60 to spend on clothes. She wants to buy a pair of jeans for $22 dollars and spend the rest on t-shirts. Each t-shirt costs $8. Write an inequality for the number of t-shirts she can purchase. 9.* Steven has $25 dollars to spend. He spent $10.81, including tax, to buy a new DVD. He needs to save $10.00 but he wants to buy a snack. If peanuts cost $0.38 per package including tax, what is the maximum number of packages that Steven can buy? Mixed Review of the Entire Expressions and Equations Domain 1. For each of the following expressions, rewrite in simplest form A. 6(2x + 3y) – 5x – 2(3y) B. 4(x – 2y) – 3(y – 2x) + 6x 2. For each of the following expressions, rewrite in factored form A. 6x – 12y B. –4x + 6y + 2 C. 2x(3 – 4y) + 2xy – 4x (Hint: rewrite in simplest form first. Then rewrite in factored form.) 52 3. For each of the following, perform the indicated operation on the given expressions: A. (7y + 6 ) + (4y – 5) B. (7y + 6 ) – (4y – 5) C. 3(x – 4) + 4(2x – 3) D. 3(x – 4) – 4(2x – 3) E. The sum of 2x + 6y and – x – y F. The difference between 2x + 6y and – x – y G. 4. 2x + 6y subtracted from When solving the equation A. Divide both sides by –x – y , what might the steps look like? , then subtract 6 from both sides 53 B. Divide both sides by , then add 6 to both sides C. Subtract 6 from both sides, then divide both sides by D. Add 6 to both sides, then divide both sides by 5. When solving the equation 2x – 3 + 4x = 15 , what might the steps look like? A. B. C. D. Combine the like terms, then add 3 to both sides of the equation, then divide both sides by 6 Combine the like terms, then subtract 3 from both sides of the equation, then divide both sides by 6 Combine the like terms, then add 3 to both sides of the equation, then divide both sides by –2 Combine the like terms, then 3 subract both sides of the equation, then divide both sides by –2 6. When solving the equation ( ) = 15 , what might the steps look like? A. Multiply both sides by , then subtract 30 from both sides B. Divide both sides by then subtract 30 from both sides C. Subtract 30 from both sides, then divide both sides by D. Subtract 30 from both sides, then multiply both sides by 7. Solve each of the following equations by showing the algebra steps: A. 2x – 2 – 4x = –14 B. ( ) = 15 8. In my last trip to CVS, I bought 4 bags of chips that cost $2.25 each and a magazine. My total before tax was $13.75. What was the cost of the magazine? 54 9. In Mr. Unson’s last trip to the Gap, he bought a pair of pants for $40 and 3 shirts. He spent a total of $130 for the entire purchase. What was the cost of each of the shirts if they were each the same price? 10. What equation can be written to find the number of textbooks, t, you can buy if the cost of each textbook is $65 and the math department is giving you m dollars to spend? A. m = 65 + t B. t = 65 + m C. m = 65t D. t = 65m 11. Mrs. Nuffer paid $165, before tax, for her recent Amazon online order. She ordered v number of $22.50 videos and they charged her a one time charge of $7.50 for shipping. Which of the following equations could be used to find the number of videos, v, she purchased? A. B. C. D. (22.5 + 7.5)v = 165 (22.5 – 7.5)v = 165 22.5v – 7.5 = 165 165 – 7.5 = 22.5v 12. Mr. Welsh’s daughter measured 53 inches at her last checkup at the pediatrician’s office. 6 months ago when she was measured she was inches. What was her average growth per month? 13. Greg wants to work overtime to pay for a new cell phone. He needs to earn at least $675 to pay for the phone he wants. His job will pay him $65 for the first four hours of overtime and $18 for each additional hour of overtime he works. Write an inequality and solve it to find how many hours of additional overtime Greg must work to have enough money to buy the phone he wants. 55 14. Part A: John went to Target and bought his supplies from the school supply list. He bought 4 notebooks for $4.50 each and 5 1-inch binders at “x” dollars each. The total before tax was $29.25. Write an equation and solve it to find the cost of each binder. Part B: When John made his school supply purchase, Target gave him a $5.00 gift card to use towards his next purchase. He returned to Target the following week and bought more notebooks for the same price he paid the week before. If “y” is the number of notebooks he purchased on this second trip and he spent more than $9, write an inequality and solve it to find the least number of notebooks he bought on the second trip. 15. A framed picture 36 inches wide and 40 inches high is shown in the diagram below. 56 The picture will be hung on a wall where the distance from the floor to ceiling is 10 feet. The center of the picture must be 7 feet from the floor. Determine the distance from the ceiling to the top of the picture frame. Show your work 16. Mary centered 4 identical mirrors horizontally on a wall that is inches wide. Each mirror is 15 inches wide and she left a space of 4 inches between each mirror. To the nearest inch what is the distance from the end of the last mirror to the end of the wall? (Drawing a picture would be helpful) 17. The perimeter of a square is 36 inches. What is the length of each side? 57 18. The perimeter of a square is 8x + 16 inches. In terms of x, what is the length of each side? 19. The perimeter of a square is 20 + 15x inches. In terms of x, what is the length of each side? 20. The perimeter of a square is 12x + 10 inches. In terms of x, what is the length of each side? 21. Mrs. DeMarco took a taxi while visiting NYC. She paid an initial charge of $8 plus $4 per mile. She paid a total of $32. Write an equation and solve it to find how many miles, m, she travelled. 22. The Caluri family has only $250 to spend at a water park. They must pay for parking, $15, and the cost of admission for each family member. Admission is $25.75 per person. Write and solve an equation that can be used to find the number of family members who can come to the water park. 23. Kyle bought x tee shirts for $10.75 each and y shorts for $24.99 each. The tax was 7.75%. Part A Explain why you can express the total cost as 1.0775(10.75x + 24.99y) 58 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Part B Can you also use 10.75x(0.0775) + 24.99y(0.0775) Explain why or why not ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ 24. What is the product of 4 and ( x + 2.4) in simplest form? 25. Mr Welsh is buying a cover for his rectangular pool. The total cost of the cover he needs is $975.85. What’s the cost of the cover per square foot? 26. The following is a diagram of Mrs Wilson’s classroom. “x” represents the width of each of the doors. The perimeter of the room is 166.8 feet. What’s the width of each door? 59 27. Which of the following is equivalent to ( A. ) B. ( ) C. ( ) D. ( ) –6(2y – 4) – 5y + 3 28. Simplify Review for Quizzes and Tests Math 7 Warm Up for Quiz List the properties used to extend each expression: ( ) ( ( ( ) ) ) 60 Math 7 Honors Warm Up for Quiz List the properties used to extend each expression: ( ) ( ) ( ) ( ) _____________________________________________________________ ____________________________________________________________ ___________________________________________________________________ 61 9 ___________________________________________________________________ Combine Like Terms: Factor Using GCF: -3 ( 2x 24aw – 8az + 5 ) – 6x – 5 Rational or Irrational? √ Change 0.12777… ̅̅̅̅ into a fraction using the algebra steps: Math 7 Exp & Equat Test Review Directions: Choose the best answer to each of the following: Show all work for partial credit. 1. In the following expression, what is the “constant”? A] -5 2. In the expression A] coefficient B] x C] 21 , the B] variable D] No constant represents the: C] constant D] None of these 62 3. Which of the following shows like terms? A] -4p , -4r B] -4p , 2p C] 4p , D] None of these B] 12n C] -6n D] None of these B] 375x C] -10x D] None of these B] 5n – 2 C] -5 D] None of these B] -10y + 40 C] -10y – 6 D] None of these 4. Simplify: A] 27n 5. Simplify: A] 40x 6. Simplify: ( ) A] 5n – 10 7. Simplify: ( A] -10y – 40 8. ) Which of the following illustrates the distributive property? A] B] ( ) C] ( ) D] None of these 63 9. ( Simplify: ) A] B] ( 10. Simplify: B] ( ) A] 12. Simplify: ) ( C] D] None of these C] D] None of these C] D] None of these ) A] 14. Subtract: ( D] None of these ) B] ) C] ) ( A] 13. Add: ( ( B] ( D] None of these ) A] 11. Simplify: C] B] ) ( ) 64 A] B] 15. Subtract: ( ) A] ( C] D] None of these C] D] None of these ) B] 16. Find the greatest common factor (GCF) of the following: A] 1 B] 9 C] 6 D] None of these B] 8c C] 12cd D] None of these B] C] 17. Find the GCF of the following: A] 32cd 18. Factor using the GCF: A] ( ) ( ) ] None of these Math 7A Expressions & Equations Practice 1. Fill in the blank for each part of the expression: 2. Name the property illustrated by: _____________________________________ 65 3. Write an equation demonstrating the multiplicative inverse property: ______________________________ 4. Name the terms in the polynomial: -5n + 2 + 7p. _____________________________________ Simplify: 5. ____________ 6. ____________ 7. ____________ 8. ____________ ( ) 9. ____________ Write “six less than twice a number f” as an expression. 10. ___________ Find the difference when is subtracted from 11. ___________ Find the difference when is subtracted by . . Solve for x: 12. __________ 13. __________ Solve for x: 14. __________ 66 15. __________ 16. __________ 17. __________ 18. __________ – ( ) 19. __________ 20. __________ Solve for x: 21. __________ 67 22. __________ 23. __________ ( ) ( ) Use an equation to solve each of the following: 24. __________ A taxi charges a $3 surcharge, plus $2 per mile. If your bill was $23, how many miles were you driven? 25. __________ You had $60, then you bought 4 bags (each the same price). You now have $28 remaining. How much was each bag? Math 7H Expressions & Equations Practice 1. Fill in the blank for each part of the expression: 68 ( 2. Name the property illustrated by: _____________________________________ ) 3. Write an equation demonstrating the multiplicative identity property: ____________________________ 4. Name the terms in the polynomial: 8g – 7 – 2x. _____________________________________ Simplify: 5. ____________ 6. ____________ 7. ____________ 8. ____________ ( ) 9. ____________ Write “four less than three times a number n” as an expression. 10. ___________ Find the difference when is subtracted from 11. ___________ Find the difference when is subtracted by . . Solve for x: 12. __________ 13. __________ Solve for x: 69 14. __________ 15. __________ 16. __________ 17. __________ 18. __________ – ( ) 19. __________ 20. __________ ( ) 70 Solve for x; then graph the solution 21. __________ 22. __________ 23. __________ ( ) Name the inequality graphed below: Use an equation to solve each of the following: 24. __________ A taxi charges an $8 surcharge, plus $6 per mile. If your bill was $56, how many miles were you driven? 25. __________ You had $20 in your pocket, then you bought 5 sodas (each the same price). You now have $13.25 remaining. How much was each soda? Review of all Topics So Far 71 Domain 1 – Ratios and Proportional Relationships MUST SHOW ALL WORK FOR CREDIT _______1. A recipe calls for cups of flour for every cup of sugar used. How many cups of flour are are needed for each cup of sugar used? _______2. Sara can run mile in minutes. What is Sara’s unit rate of speed? _______3. Ann has a coin that was worth $24 in 2014. It increased in value by 45% in 2015. It then decreased by 20% in 2016. What is the value of Ann’s coin in in 2016? _______4. What is the unit rate of speed for the car in the graph? _______5. Which table or tables show pairs of values that are not in a proportional relationship? 72 A. x y 1 1.5 2 3 3 4.5 4 6 x y 1 5 2 10 3 15 4 20 x y 1 2 2 4 3 6 4 8 x y 1 2 3 4 B. C. D. ________6. What is the constant of proportionality for the proportional relationship represented by the equation y = 4.2x 7. For the following situation first write the complex fraction you would use and then compute the unit rate. Joe traveled miles per hour. Complex Fraction: _______________ Unit Rate __________________________ 8. Write an equation for the following proportional relationship. X 1 2 3 4 Y 3.5 7 10.5 14 Equation is _________________________________ 73 9. Write an equation for the following proportional relationship. Equation is _________________________ 10. Write the constant of proportionality for the following table. x y 1 4.5 2 9 3 13.5 4 18 5 22.5 6 27 The constant is ___________________ 11. The graph shows the distance that a bus driver drives in a week. Does this graph show a proportional relationship? If so, what do the following points the following points represent in this situation: (1, 50) and (2, 100) ? Explain your reasoning. _________________________________________________ _________________________________________________ 74 12. The mapping diagram shows that there is a proportional relationship between the x values and the y values. Identify the constant of proportionality from the diagram. Then write an equation to represent the relationship shown by the diagram. Does your equation also show the constant of proportionality? Explain _______________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 13. At Best Buy, a TV was marked up 70% of what it cost Best Buy. Best Buy’s cost was $270. A store employee gets a 30% discount on anything he purchases on the marked up prices. How much does the employee pay for the TV? Employee’s Pays ________________ 14. Find the Simple Interest on a loan of $65,000 at % for 4 months. Then find the balance at the end of the months. Use I = PxRxT (Be careful it says 4 MONTHS) 75 Interest ___________________ Balance ____________________ 15. The Game Stop usually sells a game for $27.50 each. They are having a 35% off sale. Customers must pay an additional 7.75% tax on the sale price of the game. What will be the total cost of the game, including tax, rounded to the nearest penny? Final Cost___________________ 16. The average temperature in July was 82 degrees. The average temperature in December was 38 degrees. What is the percent of change to the nearest tenth of a percent? Percent of Change__________________ 17. A store sells books for $10.99. That price includes the $1.42 tax. What was the store’s percent of tax to the nearest tenth of a percent? Percent of Tax is _______________ 76 Domain #2 - The Number System SHOW ALL WORK 1. Al wanted to put a railing next to his stairs. The railing needs to be feet long. If he has 2 pieces of wood that are each feet long, how much longer must the third piece be? Write your answer in simplest form. 2. Kim is baking a batch of cookies. She needs cups of sugar. If she only has cups, how much more sugar does Kim need? Write your answer in simplest form. 3. Todd needs to make 3 steps for his deck. One step needs to be feet long and one step needs to be feet long and the third needs to be feet long. If he has a board of wood that is 20 feet long, how much more Todd need to make the steps? Write your answer in simplest form. 4. While doing her math homework, Nicole wrote the following sentence in her notebook: +( ) = ( ) ( ) Which property did she use? 5. Sheila is studying the properties of numbers. Her math teacher wrote this expression on the board: + 0 = which property does this expression illustrate? 6. Samantha is practicing for the long jump for the upcoming track and field meet. Her first jump measured feet. Her second jump was feet. How much longer was the second jump? 77 7. Tammy and her two best friends are making two batches of chocolate chip cookies. The recipe for one batch of cookies is below: cups of all purpose flour cup of sugar cup of packed brown sugar 1 cup (2 sticks) of butter softened 1 teaspoon baking soda 1 teaspoon salt 1 teaspoon vanilla extract 2 large eggs 2 cups (12-oz pkg.) chocolate chips 1 cup chopped nuts (optional) What is the total amount of flour, sugar, brown sugar and chocolate chips needed to make two batches of cookies? 8. Last week, Holly bought three five-pound bags of apples and a four-pound bag of cherries. She made an apple pie and cherry pie for the bake sale. The apple pie called for pounds of apples, and the cherry called for pounds of cherries. How many pounds of apples and how many pounds of cherries were left after she made the pies? 9. Pet Smart’s brand dog food comes in two sizes: pounds for the large bag and for the smaller size. How many pounds of dog food did Bailey purchase if she bought three large bags and two small bag of Pet Smart’s dog food? 10. Zach loves winter sports. A typical day at the slopes for him includes practicing snowboard78 ing for hours, downhill skiing for hours, and then snow tubing for an hour and a quarter. How much time does Zach spend on the slopes on a typical day? 11. Joyce is feet tall. She is at the amusement park with her family and they are about to ride the newest roller coaster. When they get to the front of the line, there is a sign that says you must be at least feet tall to ride the coaster. How much taller than the required height is Joyce? 12. Bernie spends three hours a day on homework. If he spends of an hour on Science, of an hour on Social Studies, and of an hour on ELA, how much time does Bernie spend on math? 13. Use long division to express each of the following fractions as decimals: A. B. C. 14. Cindy wants to make orders of tacos. Each taco needs ounces of cheese. How many ounces of cheese will she need? Express your answer in simplest form. 15. There are 36 students in Ms. Jayne’s third period class. her class. How many students received an A? of the students received an A in 79 16. If there are 30 days in June and Steve only worked of the days, how many days did Steve work? Express your answer in simplest form. 17. Andy pitched innings a game for 15 games. How many innings did Andy pitch? 18. The area of a rectangle is length x width. Sammy’s swimming pool is shaped like a rectangle. The length of the pool is feet. The width is feet. What is the area of Sammy’s swimming pool? 19. There are of a pound of grapes left after a picnic. Cody had enough and wants to split what is left evenly among his three friends. How much will each friend get? Express your answer in simplest form. 20. Gino has a stick of pepperoni that is inches long. He wants to cut inch pieces to put on his large pizza. How many pieces can Gino get from his stick of pepperoni? 21. The area of a rectangle is length x width. What is the length of a rectangular pool table that has a width of feet and an area of square feet? Express your answer in simplest form. 22. Find the numerical value for each of the following: A. | | B. – |– | C. –(–2) 80 D. the additive inverse of E. the multiplicative inverse of F. |( ( G. | ( ) ) (– )| ) | 23. Use the number line to answer the question. Which statement best models the number line? A. | | = 3 B. | |=3 C. | | = –3 D. –3 = 3 24. Write the following subtraction problems as addition problems and find the answer A. –6 – 5 B. –2 – (–1) C. 10 – (–10) 25. Ann wrote four statements using absolute value. Which of Ann’s statements is wrong? A. | | = –8 B. | | = 5 C. | |=3 D. – | | = –7 81 26. One scuba diver descended 15 meters below the surface of a lake. Another diver descended 8 meters below the surface. At the same time, a seagull was flying 2 meters above the lake’s surface, and another seagull was flying 10 meters above the surface. Which situation has the greatest absolute value in relation to the surface of the lake? (Hint: The surface of the lake is 0 feet) A. B. C. D. The scuba diver that is 15 meters below the lake’s surface The scuba diver that is 8 meters below the lake’s surface The seagull that is 2 meters above the lake’s surface The seagull that is 10 meters above the lake’s surface 27. A model rocket was launched from the ground and shot 150 feet straight up. It then fell back down to the ground and landed in the same place from which it was launched. Which expression shows how far the rocket traveled? A. | |–| | B. | |–| | C. | |+| | D. | | + (–| |) You may use your calculator for this part but must SHOW ALL WORK 28. Sue bought a 3 pounds of flour for cupcake wars at school to share with her friends. She used pounds of the flour and divided the remaining equally among her 4 friends. The expression below represents the number of pounds of flour Sue gave to each of her friends. ( ) A. Use the distributive property and rewrite the expression. Explain how this change would make the expression easier to evaluate. _____________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ B. Evaluate the expression you wrote in Part A. How many pounds of flour did Sue give to each of her friends? 82 Answer__________ 29. Evaluate: ( ) Show your work Answer___________ 30. John has 9 feet of wood to make new stairs for his deck. He needs step. feet for each A. Set up a complex fraction that represents the number of steps John can make. Complex Fraction________________ B. Simplify your complex fraction from Part A to solve, and express your answer as a mixed number. Mixed Number __________ 83 C. Explain what the whole number and fractional parts of the mixed number answer represent in the context of the situation described. ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 31. A number line is shown below. A. Plot the point on the given number line B. What is the opposite of the number ? ______________ C. Using the number line and the definition of additive inverse to explain why the number 3 and its opposite are additive inverses. ____________________________________________________________________ _____________________________________________________________________ 32. Point A is shown on the number line below. 84 What is the additive inverse of the number represented by point A? 33. Evaluate ─12 + 12 Evaluate 25 – 50 34. If a, b, c, and d are non-zero integers, which of the following is equal to A. B. ∙ ? C. 35. A number line is shown below. Which of the following expressions represents the distance between ─2 and 3 on the number line? A. |-2+3| B. |2 - 3| C. |-3 – (-2)| 36. Divide: ÷ 37. Simplify: ÷ 14 SHOW YOUR WORK D. |-2 -3| SHOW YOUR WORK 85 38. What is the value of 39. Evaluate ∙( ) 8 ÷ 0 ? SHOW YOUR WORK 40. Ann simplified the expression shown below as follows. –15 ─ 25 ─ (– 4) + (─1) ─40 ─ (–4) + (─1) ─36 + (─1) ─37 Step 1 Step 2 Step 3 In which Step was Ann’s mistake? Correct the problem starting on the line where her mistake occurred and complete the problem to get the correct answer. 41. On the number line shown below, point A has a value of 4 What number must be added to 4 to get a sum of 0 ? 86 42. Evaluate: – + ( ) + HINT: You might want to look at it first and think about your properties. MUST SHOW YOUR WORK 43. The energy consumption of an appliance is measured in kilowatt-hours (kWh) and is the product of the kilowatts per hour the appliance uses and the number of hours it uses energy. The Unson family’s washer uses 2 kilowatts per hour. If Mr. Unson runs the washer for MUST SHOW YOUR WORK. 44. hours, how much energy will the washer use? Which of the following situations could be represented by the equation shown below? + ( ) = –3 A. The temperature one morning was 4 degrees Celsius. It decreased by 8 degrees Celsius during the day to reach a low of Celsius. B. Joe studied for 4 hours, starting at 8:00 PM, so he still needs to study for 3 1/2 more hours. 87 C. Mary spent dollars on a sandwich. She had 8 dollars, so she has 3 1/2 dollars left. D. Louis is 4 miles away from finishing a marathon. He is running at a speed of 8 miles per hour, so he will finish in 3 1/2 hours. 45. A scuba diver finds some interesting sea creatures at 5 feet below sea level, represented by the number – , and another interesting group of sea creatures at at feet below sea level represented by the number . What is the distance between these two points below sea level? MUST SHOW YOUR WORK Domain #3 – Expressions and Equations 1. Distrbute 3(x + 2) 2. Simplify –2(m + 3) 3. Combine like terms to simplify 3x + 5 + 2x 4. Combine like terms to simplify the expression 4m + 2n – m – n 5. Simplify 6t + t 6. Which expression is equivalent to 4x + 3 – 2x + 5 A. 2x + 8 B. 7x – 3 C. 6x + 8 D. 12x + 8 7. Which expression correctly simplifies 4(x – 5) A. 4x + 5 B. 4x + 20 C. 4x – 5 D. 4x – 20 88 8. Simplify 5x + x 9. Simplify 3r + 2r + r 10. Simplify –2(x + 3) 11. Simplify –7(2x – 4) 12. Simplify 4(4x – 3y + 6) 13. Combine like terms and simplify 6x + 4 + 5x 14. Combine like terms and simplify 16y + 9 + 4 + 4y 15. Combine like terms and simplify 4r – 6 – 2r + 5 16. Combine like terms and simplify 13p + 7 – 14p – 8 17. Which expression is equivalent to 4s + 3 – 3 – s A. 3s + 6 B. 4s + 0 C. 3s D. 3s + 1 18. Which expression is equivalent to 6w + 5 – 3w + 7 A. 9w + 2 B. 3w + 12 C. 11w – 4 D. 11w + 4 19. Which expression is equivalent to 3d – d + 4 + d A. 4d + 4 B. 4d – 4 C. 3d – 4 D. 3d + 4 89 20. Which expression is equivalent to –u + 5 – 3u + 4 A. –4u + 9 B. –3u + 9 C. –2u + 9 D. 2u + 9 21. Simplify 6a + 12b + 5a – 6b + 2 22. Simplify the expression 5g + 5 – 2g + g + 5h 23. Simplify the expression 21q – 4v – 8q + q – v 24. The sum of 2xy and 17xy is A. 19 B. 19xy C. 34xy D. 15xy 25. The difference between 25ab and 37ab is A. 12ab B. –12 C. 12 D. –12ab 26. The difference between –7xy and –9xy is A. –16xy B. –16 C. 2xy D. –12xy 27. The sum of –xy and –9xy 28. The sum of –81bc and 7bc 29. The sum of 112cx, 3c, –12cx and 7c 30. 16x + 3xy – 8x – 7xy is equivalent to 90 A. 19xy – 15x B. –8x – 4xy C. 8x – 4xy D. 8x + 4xy 31. –7b – 9ac + 4b + 12ac is equivalent to A. –3ac + 3b B. 3ac – 3b 32. Simplify the following expression: C. ac – b D. 3b – 3ac 2(xy + 2x) – 3xy 33. Simplify the following expression: –2(3x – 3xy) + xy 34. Simplify the following expression: x(5y – 3) + 6x – 2xy 35. Simplify the expression 2x(y – 1) – 7xy + 2x 36. For each of the following solve for x by showing the appropriate equation solving steps: A. 3x – 4 = 23 B. 5 + C. –4x – 8 = –16 D. 3.2 + 1.2 = 4.56 E. F. 91 G. –6x + 21 = 51 H. –x + 25 = 100 I. 6 + 3m = 24 J. x + 2x = 90 K. 5t – 2 – t = 14 L. –3(x – 2) + 1 = 28 M. 6(x – 2) – 4x = 16 N. 9x – 4(x – 3) = 72 O. 15x – 3(3x + 4) = 6 37. Max is saving $10 a month for a summer vacation. If he also has $50 from his grandmother, how many months will Max need to save to have a total of $200? Choose the equation that represents this situation. 92 A. B. C. D. 50m + 10 = 200 50m – 10 = 200 10m – 50 = 200 10m + 50 = 200 38. When you solve 4(x + 2) = 22, you must simplify the equation first. What would the first step look like? A. B. C. D. 4x + 8 = 22 4x + 2 = 22 x + 8 = 22 x + 2 = 22 39. When solving equations, sometimes you have to simplify the equation first. When solving 3x + 2 + x – 6 = 25, what might your first step look like? A. 3x + 8 = 25 B. 3x – 4 = 25 C. 4x + 8 = 25 D. 4x – 4 = 25 40. James bought a sweater that cost $15.99 and some socks that cost $0.99 per pair. The total cost was $21.93. Write an equation and solve it to find how many pairs of socks James bought. 41. Yesterday, Mary mailed her wedding invitations. Today she mailed 72 more invitations than she did yesterday. If Mary mailed 200 invitations in all, how many invitations did she mail yesterday? 93 42. John bought 3 cars in the last year for a total of $30,000. He paid the same amount for each of the first 2 cars, and got the 3rd one for twice the amount of the 1st car. Which equation can be used to find out how much John paid for each car? A. B. C. D. x(30,000) = 2x x + 2x = 30,000 30,000x = x + 2x x + x + 2x = 30,000 43. There are 3 flights that depart from New York and arrive in London. Combined, the 3 planes hold 1,200 people. If there are 405 seats in the first plane, and 345 more seats in the third plane than in the second, how many seats are in the second plane? A. B. C. D. 405 + 1200 + x + x = 345 405 + x + x + 345 = 1200 405 + x + 345X = 1200 405(2x) + 345 = 1200 44. If you add 125 to of a number, the result is 450. Which equation would help find the original number? A. ( ) B. C. D. 45. Yesterday Bill bought some bottles of soda for a party he is hosting. Today he bought 8 times as many bottles of soda as he did yesterday. Bill has 27 bottles 94 in all. Which equation would help you to find out how many bottles of soda Bill bought yesterday? A. B. C. D. x + 8x = 27 8x – x = 27 8x(x) = 27 8x ÷ x 27 46. Which inequality represents the translation of the following sentence? “Twice a number increased by four is more than 7” A. B. C. D. 2x + 4 < 7 2x – 4 > 7 2x + 4 > 7 2x + 4 ≤ 7 47. Which inequality represents the translation of the following sentence? “Half a number is less than four more than the same number” A. C. 2x > x + 4 B. D. 48. Which inequality represents the translation of the following sentence? “The product of a number and three is less than or equal to that number decreased by five” A. B. C. D. 3x ≤ 3x ≥ 3+x 3x ≤ 5–x x–5 ≤ x–5 x–5 49. Which inequality represents the translation of the following sentence? “Four more than two times a number is less than or equal to five” A. 4 + 2x > 5 B. 2x + 4 < 5 95 C. 4 + 2x ≥ 5 D. 2x + 4 ≤ 5 50. Which inequality represents the translation of the following sentence? “Twelve less than a number is greater than that same number divided by two” A. x – 12 ≥ 2x B. 12 – x ≥ C. x – 12 > D. x – 12 < 51. Which inequality represents the translation of the following sentence? “The quotient of a number and 5 is less than or equal to that number increased by 1” A. 5x < x + 1 B. C. D. 52. Vincent has cut three pieces of rope to complete a science project. Two pieces are of equal length. The third piece is one-quarter the length of each of the others. He cut the three pieces from a rope 54 meters long without any rope left over. Find the number of meters in each piece. 53. A restaurant sells large and small submarine sandwiches. Rolls for the sandwiches are ordered from a baker. The roll for the large sandwich costs $0.25 and the roll 96 for a small sandwich costs $0.15. Melissa, the manager of the restaurant, ordered 130 more large rolls than small rolls. What was the greatest number of large rolls she received if she spent less than $63. 54. Two numbers are in the ratio 5:6. If the sum of the numbers is 66, find the value of the larger number. A. 6 B. 24 C. 30 D. 36 55. At a concert, $720 was collected for hot dogs, hamburgers, and soft drinks. All three items sold for $1.00 each. Twice as many hot dogs were sold as hamburgers. Three times as many soft drinks were sold as hamburgers. The number of soft drinks sold was A. 120 B. 240 C. 360 D. 480 56. A boy got 50% of the questions on a test correct. If he had 10 questions correct out of the first 12, and of the remaining questions correct, how many questions were on the test? A. B. C. D. 16 24 26 28 57. A doughnut shop charges $0.70 for each doughnut and $0.30 for a carryout box. Shirley has $5.00 to spend. At most, how many doughnuts can she buy if she also 97 wants them in one carryout box? 58. There are 28 students in a mathematics class. If the guidance office, of the students are called to of the remaining students are called to the nurse, and finally, of those left go to the library, how many students remain in the classroom? 59. Peter begins his kindergarten year able to spell 10 words. He is going to learn to spell 2 new words every day. Which inequality can be used to determine how many days, d, it takes Peter to be able to spell at least 75 words? A. B. C. D. 10 + 2d ≥ 75 10 + 2d ≤ 75 2 + 10d ≥ 75 2 + 10d ≤ 75 60. Tamara has a cell phone plan that charges $0.07 per minute plus a monthly fee of $19.00. She budgets $29.50 per month for total cell phone expenses without taxes. What is the maximum number of minutes Tamara could use her phone each month in order to stay within her budget? A. B. C. D. 150 271 421 692 98 61. The table below represents the number of hours a student worked and the amount of money the student earned. Number of Hours (h) 8 15 19 30 Dollars Earned (d) $50.00 $93.75 $118.75 $187.50 Which equation correctly represents the number of dollars, d, earned in terms of the number of hours, h, worked? A. B. C. D. h = 6.25d d = 6.25h h = 50 + 6.25d d = 50 + 6.25h 62. An online music club has a one-time registration fee of $13.95 and charges $0.49 to buy each song. If Emma has $50.00 to join the club and buy songs, what is the maximum number of songs she can buy? A. B. C. D. 73 74 130 131 63. The ninth grade class at a local high school needs to purchase a park permit for $250.00 for their upcoming class picnic. Each ninth grader attending the picnic pays $0.75. Each guest pays $1.25. If 200 ninth graders attend the picnic, which inequality can be used to determine the number of guests, x, needed to cover the cost of the permit? A. B. C. D. 0.75x – (1.25)(200) 0.75x + (1.25)(200) (0.75)(200) – 1.25x (0.75)(200) + 1.25x ≥ ≥ ≥ ≥ 250.00 250.00 250.00 250.00 99 DIRECTIONS: Find the mistake in each question; indicate where the first mistake occurs. Describe the mistake and the correction. Expand each expression #64 3(x + 9) 3x + 9 #65 6(x + 2) 6x + 6(2) 6x + 12 18x #66 5(x – 6) 5(-5) -25 100 Factor each expression #67 4n + 8 4(n + 2) 4(3) 12 #68 20n – 4ny 2n(10 – 2y) #69 -18n + 8 2(-9n – 4) Simplify #70 #71 #72 Enrichment ALL OF THE FOLLOWING ARE TAKEN FROM NORTH CAROLINA’S CCSS (They had a footnote acknowledging that some of the problems and graphics were taken from the Arizona Department of Education) 8.EE.1 1. 101 2. 3. 4. 5. ( )( 6. ( ) 7. ( ) ) ( ) ( ) 8.EE.2 1. difference between ( 2. Understanding ( ) 3. Solve = 4. Solve = 27 5. Solve = ) and understanding that √ = ±4 and √ 6. What is the side length of a square with an area of 49 8.EE.3 1. Write 75,000,000,000 in scientific notation 2. Write 0.0000429 in scientific notation 3. Express 2.45 x in standard form 102 4. How much larger is 6 x compared to 2 x 5. Which is the larger value: 2 x or 9 x 8.EE.4 1. Problems using scientific calculators using E or EE (scientific notation), *(multiplication), and ^(exponent) symbols 2. In July 2010 there were approximately 500 million facebook users. In July 2011 there were approximately 750 million facebook users. How many more users were there in 2011. Write your answer in scientific notation. 103 3. (6.45 x )(3.2 x ) 4. 5. (0.0025)(5.2 x ) 6. The speed of light is 3 x meters/second. If the sun is 1.5 x meters from earth, how many seconds does it take light to reach the earth? Express your answer in scientific notation. 8.EE.5 1. Compare the scenarios to determine which represents a greater speed. Explain your choice including a written description of each scenario. Be sure to include the unit rates in your explanation. 104 Scenario 1: Scenario 2: Traveling Time y = 55x (shows a graph with time as the horizontal axis, distance (miles) as the vertical axis and the line drawn includes the points (1,60) and (4, 240) x is time in hours y is distance in miles 8.EE.6 1. (Shows coordinate plane with two triangles shaded in) The triangle between A and B has a vertical height of 2 and a horizontal length of 3. The triangle between B and C has a vertical height of 4 and a horizontal length of 6. The simplified ratio of the vertical height to the horizontal length of both triangles is 2 to 3, which also represents a slope of for the line, indicating that the triangles are similar. 2. Given an equation in slope-intercept form, students graph the line a. b. c. d. 3. Students write equations in the form y = mx for lines going through the origin and recognize that m is the slope of the line See hand out 4. Write an equation of a line that is graphed that goes through the origin. Write an equation of a line that is graphed that does not go through the origin understanding y = mx + b 105 8.EE.7 1. Solve 10x – 23 = 29 – 3x 2. Equations that have no solution because the variables cancel out: –x + 7 – 6x = 19 – 7x 3. Equations with infinitely many solutions when both sides are the same: ( ) ( ) 4. Two more than a certain number is 15 less than twice the number. Find the number. 106 8.EE.8 1. Plant A and Plant B are on different watering schedules. This affects their rate of growth. Compare the growth of the two plants to determine when their heights will be the same. (Given a table, then based on the table they make a graph. Then they write an equation for each representing the growth rate of Plant A and the growth rate of Plant B) Determine at what week will the plants have the same height based on your equations. Write and solve an equation to find the week. 2. Victor is half as old as Maria. The sum of their ages is 54. How old is Victor? Multiplication and Division Property of Inequalities 107 If you multiply or divide each side of an inequality by a positive number, leave the inequality symbol unchanged. If you multiply or divide each side of an inequality by a negative number, reverse the inequality symbol. Example: When you multiply or divide by a negative in the last step you have to switch the sign around -4x > 20 -4x > 20 -4 -4 x < -5 When you multiply or divide by a negative in the last step you have to switch the sign around. x<9 -3 -3x < 9(-3) -3 x > -27 **But 2x > -10 you are not going to switch the sign around Practice: Solve and graph each of the following 108 1. –3x + 9 ≥ –21 2. x + 15 > 23 -8 4. 55 > –7x + 6 5. –25 + x ≤ 50 2 7. 10 + x > 12 –9 8. –6x – 18 ≥ 36 3. 2x + 5 > -49 6. x – 22 < 48 2 9. –2x + 12 > 8 109 Multiple Step Equation Solving with Variables on Both Sides 1. 12x - 3 = 8x + 21 2. 5(2x + 3) + 2x = x + 48 3. 3(2x + 4) = 2(2x - 20) 4. 4(x – 2) = 3x + 1 25y + 10 - 15y + 20 = 100 5. 10(y + 6) + 3y = 86 6. 7. 14x + 33 - 3x - 11 = 33 8. 3(y + 5) + 2y + 10 = 50 9. 3(y – 2) + 6 = 90 10. 2y + 17 + 3 + 3y = 100 110 11. 4(x + 3) = 2x - 24 13. 12x – 4 = 2x + 6 12. 2x + 3(x + 1) = x + 31 14. 3(x + 2) + 5 = 2x + 44 15. 2(x – 5) + 8 = 16 16. 4(x – 2) = 3(x + 4) 17. 10y + 3 = 6y + 27 18. -2y + 6 = 4y - 36 19. x(6 – 3) + 12 = x + 20 20. 2(y + 2) = 5(y – 4) 111