Name Date Class. Solving Equations by Adding or Subtracting

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1 Name Date Class 2-1 Problem Solving Solving Equations by Adding or Subtracting Write the correct answer. 1. Michelle withdrew $120 from her bank account. She now has $3345 in her account. Write and solve an equation to find how much money m was in her account before she made the withdrawal. _ 2. Max lost 23 pounds while on a diet. He now weighs 184 pounds. Write and solve an equation to find his initial weight w. _ 3. Earth takes 365 days to orbit the Sun. Mars takes 687 days. Write and solve an equation to find how many more days d Mars takes than Earth to orbit the Sun. 4. In 1990, 53.4% of commuters took public transportation in New York City, which was 19.9% greater than the percentage in San Francisco. Write and solve an equation to find what percentage of commuters p took public transportation in San Francisco. Use the circle graph below to answer questions 5 7. Select the best answer. The circle graph shows the colors for SUVs as percents of the total number of SUVs manufactured in 2000 in North America. 5. The percent of silver SUVs increased by 7.9% between 1998 and If x% of SUVs were silver in 1998, which equation represents this relationship? A x = 14.1 C 7.9x = 14.1 B x 7.9 = 14.1 D 7.9 x = Solve the equation from problem 5. What is the value of x? F 1.8 H 7.1 G 6.2 J The sum of the percents of dark red SUVs and white SUVs was 26.3%. What was the percent of dark red SUVs? A 2.3% C 12.2% B 3.2% D 18% 2-9 Holt McDougal Algebra 1

2 Name Date Class 2-1 Challenge Rate Problems The function of an odometer is to measure the distance that a vehicle has traveled, either in miles or in kilometers. When you use an odometer for this purpose, you can answer the question, How far have I traveled? The reasoning that you used to answer the question can be applied to stating and solving new related problems. Answer the following questions. 1. You are on a trip traveling from town A to town B. When you start the trip, your odometer reading is 37,538 miles. When you get to town B, your odometer reading is 37,781. a. Write an equation relating your initial reading, distance traveled, and final reading. b. Find the distance traveled from town A to town B. 2. Let s represent the initial reading, d represent the distance traveled, and e represent the final reading. Write an addition equation relating s, d, and e. 3. Using your equation from Exercise 2, solve for the specified variable. a. s = b. d = c. e = 4. Use one of the equations that you wrote in Exercise 3 and the given values to find the value of the third variable. a. s = 27,281 and e = 28,978 b. e = 17,349 and d = 197 _ c. s = 62,979 and d = 798 d. s = 69,876 and e = 70,987 _ The water level in a storage tank is currently feet. 5. Water is drained from the tank, and the level drops feet. Find the new water level. 6. Let o represent the original water level, d represent the change in water level, and f represent the final water level. Write three equations relating o, d, and f. 7. The water level rises feet from feet. Choose an equation from Exercise 6 and find the new water level. 2-8 Holt McDougal Algebra 1

3 Name Date Class 2-1 Review for Mastery Solving Equations by Adding or Subtracting Use counters to model solving equations. Solve x + 2 = 5. Using counters Using numbers x + 2 = 5 x + 2 = x + 0 = 3 x = 3 Check: Check: x + 2 = =? 5 5 =? 5 Solve the following by drawing counters. Check your answers. 1. x + 1 = = x + 2 _ Solve each equation. Check your answers. 3. x + 4 = = x x + 3 = Holt McDougal Algebra 1

4 Name Date Class 2-1 Review for Mastery Solving Equations by Adding or Subtracting continued Any addition equation can be solved by adding the opposite. If the equation involves subtraction, it helps to first rewrite the subtraction as addition. Solve x + 4 = 10. x + 4 = 10 The opposite of 4 is Add 4 to each side. x = 6 Solve 5 = x 8. 5 = x + 8 Rewrite subtraction as addition. The opposite of 8 is Add 8 to each side. 3 = x Solve x ( 6) = 2. x + 6 = x = 4 Find the opposite of this number. Find the opposite of this number. Find the opposite of this number. Rewrite subtraction as addition. The opposite of 6 is 6. Add 6 to each side. Check: x + 4 = =? =? 10 Check: 5 = x 8 5 =? =? 5 Check: x ( 6) = 2 4 ( 6) =? 2 2 =? 2 Rewrite each equation with addition. Then state the number that should be added to each side. 6. x 7 = x ( 1) = = x 2 Solve each equation. Check your answers. 9. x + 10 = = x x ( 5) = Holt McDougal Algebra 1

5 Name Date Class 2-1 Practice C Solving Equations by Adding or Subtracting Solve each equation. Check your answers. 1. d 17 = 4 2. f = = n x = 8 5. h = y = 23 Write an equation for each relationship. Then solve the equation. 7. A number increased by 12 is equal to Seven less than a number is 3. _ 9. The difference of a number and 6 is The sum of a number and 4.7 is 8.1. _ 11. Terrell s plant grew inches in one month, and it is now inches tall. Write and solve an equation to find the height of the plant at the start of the month. Show that your answer is reasonable. 12. After installing new software, the available memory on Joy s computer dropped by 36.8 MB, leaving her with MB of memory. Write and solve an equation to determine the amount of available memory on Joy s computer before installing the new software. Show that your answer is reasonable. 13. For all sanctioned USA swim meets in Indiana in 2004, the fastest time for the 50-meter freestyle in the girls age group was 28.3 seconds. This time was 0.43 seconds faster than the fastest time in Write and solve an equation to determine the fastest time for this swim event in Indiana in Show that your answer is reasonable. 2-5 Holt McDougal Algebra 1

6 Name Date Class 2-1 Practice B Solving Equations by Adding or Subtracting Solve each equation. Check your answers. 1. g 7 = t + 4 = = m 7 4. x = n 3 8 = p 1 3 = k = = w = r y 57 = b = a + 15 = Marietta was given a raise of $0.75 an hour, which brought her hourly wage to $ Write and solve an equation to determine Marietta s hourly wage before her raise. Show that your answer is reasonable. 14. Brad grew inches this year and is now inches tall. Write and solve an equation to find Brad s height at the start of the year. Show that your answer is reasonable. 15. Heather finished a race in 58.4 seconds, which was 2.6 seconds less than her practice time. Write and solve an equation to find Heather s practice time. Show that your answer is reasonable. 16. The radius of Earth is km, which is km longer than the radius of Mars. Write and solve an equation to determine the radius of Mars. Show that your answer is reasonable. 2-4 Holt McDougal Algebra 1

7 Name Date Class 2-1 Practice A Solving Equations by Adding or Subtracting Solve each equation by using addition. Check your answers. 1. m 2 = 5 2. t 9 = p 6 = 2 4. a 4.5 = = c 8 6. y 1 5 = 2 5 Solve each equation by using subtraction. Check your answers. 7. b + 4 = w = 1 9. p + 6 = = x x + 17 = r = James took two math tests. He scored 86 points on the second test. This was 18 points higher than his score on the first test. Write and solve an equation to find the score James received on the first test. Show that your answer is reasonable. 14. The noontime temperature was 29 F. This was 4 F lower than predicted. Write and solve an equation to determine the predicted noontime temperature. Show that your answer is reasonable. 15. For two weeks, Gabrielle raised money by selling magazines. She raised $72 during the second week, which brought her total to $122. Write and solve an equation to find how much Gabrielle raised during the first week. Show that your answer is reasonable. 2-3 Holt McDougal Algebra 1

8 Name Date Class 1-7 Problem Solving Simplifying Expressions Write the correct answer. 1. An English teacher gives students 1 point for reading a magazine article, 5 points for reading a chapter of a book, and 20 points for completing an entire book. If Sue reads 4 magazine articles, 7 chapters, and completes 2 books this term, how many points will she earn? 2. A recipe for chocolate chip cookies calls for 2 1 cups of flour, 1 cup of butter, cup of brown sugar, 3 cup of sugar, 4 and 1 cup of chocolate chips. Find the total number of cups of ingredients. 3. A rectangular desktop has a length of 3(x + 2) units and a width of x 7 units. Write an expression, in simplified form, for the perimeter of the desktop. 4. Lucy is k years old. She has a sister who is three years younger than she is and another sister who is five years less than twice Lucy s age. Write an expression, in simplified form, for the sum of the three girls ages. Use the table below for questions 5 8. The table shows expected flight times to and from New York City and five other cities. The legs of each trip vary in time due to the wind. Select the best answer. 5. Find the sum of the expected outbound flight times. A 23 h B 26 h C h D h 6. If Marty plans to travel from New York to Paris and back in February, and then from New York to Rome and back in April, what will be his total flight time for both trips? F h G h H 31.0 h J 31.5 h 7. Juan s flight time to San Diego was x hours longer than expected. His flight back was y hours less than expected. Which expression shows Juan s total flight time? A 10.15xy B 5.4x 4.75y C x y D 5.4x( 4.75y) City Expected Flight Times Inbound (h) Outbound (h) Mexico City Paris San Diego Atlanta Rome Last month, Heather flew from New York to Atlanta and back twice a week for 3 weeks. What was her total flight time if there were no delays? F 12.9 h G 13.8 h H 19.8 h J 25.8 h 1-57 Holt McDougal Algebra 1

9 Name Date Class 1-7 Challenge Distributing More Than Once Sometimes it is helpful to use the Distributive Property more than once. Write the product 82(63) using the Distributive Property. Then simplify. 82(63) (80 + 2) (60 + 3) Rewrite 82 as , and rewrite 63 as (80 + 2) (60) + (80 + 2) (3) Use the Distributive Property to distribute (80 + 2). 80(60) + 2(60) + 80(3) + 2( 3) Distribute 60 through (80 + 2). Do the same for Multiply Add. Write each product using the Distributive Property. Then simplify (74) 2. 39( 92) 3. 96(98) You can also distribute several times to simplify algebraic expressions. Use the Distributive Property to simplify the product (x + 5)(x 4). (x + 5)(x 4) (x + 5)[x + ( 4)] To subtract 4, add 4. (x + 5)(x) + (x + 5)( 4) Distribute (x + 5). x(x) + 5(x) + x( 4) + 5( 4) Distribute x through (x + 5). Do the same for 4. x 2 + 5x 4x 20 Multiply. x(x) can be written as the power x 2. x 2 + x 20 Combine like terms. Use the Distributive Property to simplify each product. 4. (x + 3)(x + 8) 5. (x 2)(x + 7) 6. (a + b)(c + d) 1-56 Holt McDougal Algebra 1

10 Name Date Class 1-7 Review for Mastery Simplifying Expressions The following properties make it easier to do mental math. Property Addition Multiplication Commutative Property = = 5 2 Associative Property (3 + 4) + 5 = 3 + (4 + 5) (2 4) 10 = 2 (4 10) Distributive Property 2(5 + 9) = 2(5) + 2(9) Simplify Identify compatible numbers Use the Commutative Property to rearrange the numbers (14 + 6) Use the Associative Property to group the compatible numbers Add. 57 Simplify 5(24). 5(24) 5(20 + 4) Break apart 24 into numbers compatible with 5. 5(20) + 5(4) Distribute Multiply. 120 Add. Use the properties above to simplify each expression (32) (88) 1-54 Holt McDougal Algebra 1

11 Name Date Class 1-7 Review for Mastery Simplifying Expressions continued Terms can be combined only if they are like terms. Like terms can have different coefficients, but they must have the same variables raised to the same powers. Like Terms 4x 2 2, 7x Not Like Terms 3m, 5m 3 12y, 18y 12y, 12xy 5ab 2, ab 2 st 4, 3s 4 t Simplify 24x 3 4x 3. 24x 3 3 4x 3 20x Simplify 4(x + y) + 5x 9. 4x + 4y + 5x 9 Distribute 4. 4x + 5x + 4y 9 Subtract the coefficients only. Use the Commutative Property. 9x + 4y 9 Add the like terms 4x and 5x. 9x + 4y 9 No other terms are like terms. State whether each pair of terms are like terms. 7. 4xy and 3xy 8. 2s 2 and 5s 9. 10a and 10b If possible, simplify each expression by combining like terms st 3st y y 4y x x Simplify each expression (x + 6) y + 2(y 5) + y _ 1-55 Holt McDougal Algebra 1

12 Name Date Class 1-7 Practice C Simplifying Expressions Simplify each expression Write each product using the Distributive Property. Then simplify (12) 8. 7(57) 9. 13(103) Simplify each expression by combining like terms y + 17y p p n + 18n 15n x x + 8x m t 4t t 15t Simplify each expression. Justify each step b 2 + 5b b ( 8x + 4) (6x 24) _ Given the perimeter P, find the missing measurement. 18. _ 1-53 Holt McDougal Algebra 1

13 Name Date Class 1-7 Practice B Simplifying Expressions Simplify each expression Write each product using the Distributive Property. Then simplify (12) 8. 5( 47) 9. 4(106) Simplify each expression by combining like terms x + 27x 11. 4m + 12m 12. 6t 2 2 2t 13. 5w w 14. 4p + 7p d 3.4d Simplify each expression. Justify each step ( x + 9) + 5x d d + 18 Give an expression in simplified form for the perimeter of each figure _ 1-52 Holt McDougal Algebra 1

14 Name Date Class 1-7 Practice A Simplifying Expressions Simplify each expression Fill in the blanks to write each product using the Distributive Property (11) 8. 15(18) 9. 3(109) = 25( ) + 25(1) = 15(20) (2) = 3(100) + 3( ) = = 30 = = = 270 = Simplify each expression by combining like terms x + 18x 11. 9n 2 4n y + 18y 13. 4t 3 2 2t d 20d 15. 4r r Fill in the blanks with the property that was used for that step: Associative, Commutative, or Distributive n n (x + 5) 2x = 8 + 3n + 12n = 4x x = 8 + (3n + 12n) = 4x 2x + 20 = n Add like terms. = 2x + 20 Subtract like terms. Write and simplify an expression that represents the perimeter of each figure shown _ 1-51 Holt McDougal Algebra 1

15 Name Date Class 1-6 Problem Solving Order of Operations Write the correct answer. 1. A can of soup is in the shape of a cylinder with radius 3.8 cm and height 11 cm. What is the surface area of the can to the nearest tenth? Use 3.14 for π. (Hint: The expression 2πr 2 + 2πrh represents the surface area of a cylinder, where r is the radius and h is the height.) 2. One Boston household used the following amounts of electricity to run its heating system during the winter. Month Kilowatt-Hours Used December 1500 January 1463 February 2260 Write an expression that can be used to find the average number of kilowatt hours used. Then simplify the expression. _ 3. In a polygon with n sides, the sum of the measures of the interior angles is 180(n 2). What is the sum of the measures of the interior angles of a hexagon? _ Select the best answer. 5. Anthony had 10 packages of markers. Each package contained 8 markers. He gave his 3 best friends 2 packages each. Which expression shows how many markers he kept for himself? A B 8(10 3 2) C D 8( ) 7. Each month, Mrs. Li pays her phone company $28 for phone service, and $0.07 per minute for long distance calls. Which expression represents her bill for a month in which long distance calls totaled 4 hours? A 4[ (0.07)] B (60)(0.07) C D (4) 4. In a regular polygon with n sides, the measure of each interior angle is 180(n 2). What is the measure n of an interior angle of an octagon? _ 6. The area of the wall hanging below can be approximated by simplifying (3.14)(72 ). Which is closest to the area of the wall hanging? F sq in. G sq in. H sq in. J sq in Holt McDougal Algebra 1

16 Name Date Class 1-6 Review for Mastery Order of Operations When an expression contains more than one operation, the operations must be performed in a certain order. I. Evaluate powers (exponents). II. Perform multiplication and division in order from left to right. III. Perform addition and subtraction in order from left to right. Simplify Identify powers Evaluate Identify multiplication and division Evaluate Start at the left and perform each addition and subtraction in order Fill in the blanks to simplify each expression Simplify each expression Holt McDougal Algebra 1

17 Name Date Class 1-6 Review for Mastery Order of Operations continued Expressions can also include grouping symbols. Parentheses ( ), brackets [ ], and braces { } are the most common grouping symbols. Operations inside grouping symbols must always be done first. If there are grouping symbols inside other grouping symbols, evaluate the innermost group first. Simplify the expression 6 2 3(5 1) (5 1) Evaluate Evaluate Evaluate Add and subtract from left to right. 26 The symbols shown at right are also treated as grouping symbols. Symbol Example Absolute-value 2 3 Radical Fraction Bar Simplify each expression (8 5) (3 + 2)(4 + 3) ( ) 13. If a right triangle has legs of lengths a and b, then the length of its hypotenuse can be found using the expression a 2 + b 2. Find the length of the hypotenuse of a right triangle whose legs measure 11 cm and 14 cm. Round your final answer to the nearest tenth Holt McDougal Algebra 1

18 Name Date Class 1-6 Practice C Order of Operations Simplify each expression (12 15) 3. 3[ 2(8 13)] ( 14 2 ) Evaluate each expression for the given value(s) of the variable(s) ( b) 3 for b = x y 2 for x = 1 and y = 4 9. _ n 2 2(n + 8) for n = c for c = 2 and d = 7 c d + 3 _ 11. Translate the difference of x and the product of 8 and 6 into an algebraic expression. 12. The amount of money in a savings account after 10 years, with an initial deposit of P, annual interest rate of r, compounded quarterly, can be found using P 1+ r 40 4, assuming no other deposits or withdrawals are made. Write the steps you would take to evaluate the expression. 13. The area of a trapezoid can be found using the expression 1 2 h(b 1 + b 2 ). Find the area of the trapezoid shown Holt McDougal Algebra 1

19 Name Date Class 1-6 Practice B Order of Operations Simplify each expression (4) 3. 2[7 + 6(3 5)] 4. 7 (2 4 8) ( ) 6. 2( 13 4) Evaluate each expression for the given value of the variable y for y = (x ) for x = 8 9. (m + 6) (2 5) for m = t t for t = 10 2 Translate each word phrase into a numerical or algebraic expression. 11. the pr od uct of 6 an d the su m of 3 an d the abs olut e val ue of the differe nce of m and the qu otie nt of 18 and the sum of 2 and d Degrees Fahrenheit F can be converted to degrees Celsius C using the expression 5 (F 32). Degrees Celsius can be converted to 9 degrees Fahrenheit using the expression 9 5 C Th e h ottest recor de d d ay in Flo rid a history was 10 9 F, which occurred on June 29, 1931 in Monticello. Convert this temperature to degrees Celsius. Round your answer to the nearest tenth of a degree. 15. Th e col dest rec or ded day i n Fl ori da history was a bout 18.9 C, which occurred on February 13, 1899 in the city of Tallahassee. Convert this temperature to degrees Fahrenheit. Round your answer to the nearest tenth of a degree Holt McDougal Algebra 1

20 Name Date Class 1-6 Practice A Order of Operations Simplify each expression (10 2) (14 6) 5. 2[4(3 + 8)] ( ) 6 Evaluate each expression for the given value of x x + 2 for x = x2 2 for x = 3 _ 12. 2(x 16) 2 for x = x for x = 8 _ Translate each word phrase into a numerical expression. 14. the sum of 8 and the absolute value of the quotient of 18 and the product of 1 and the product of 7 and the difference of 8 and n 17. Find the surface area of the rectangular prism shown at right. Use the expression 2lw + 2lh + 2wh. cm Holt McDougal Algebra 1

21 Name Date Class 1-5 Problem Solving Square Roots and Real Numbers 1. Jack is building a square pen for his dog. If he wants the area of the pen to be 121 square feet, how long should he make each side of the pen? 3. The Statue of Liberty, which sits on Liberty Island in New York Harbor, is feet high, from base to torch. 12 Write all classifications that apply to : natural, whole, integer, 12 rational, terminating decimal, repeating decimal, and irrational. 2. Danny needs a square-shaped picture to cover a hole in his wall. It has to cover at least 441 square inches of wall space. What is the smallest side length the picture can have? 4. A square note card has an area of 5 in 2. Estimate the length of the side to the nearest tenth. Then write all classifications that apply to the actual side length: natural, whole, integer, rational, terminating decimal, repeating decimal, and irrational. Use the table below to answer questions 5 7. The table shows the area of four sizes of square-shaped pizzas sold at Town Pizza. Complete the table by finding the length of each side of the four pizzas. Round to the nearest tenth if needed. Select the best answer. 5. What is the length of each side of an extra large pizza? A 24 in. B 25 in. C 26 in. D 36 in. 6. Which of the following classifications applies to the length of each side of a large pizza? F natural G whole H integer J rational 7. Which of the following is NOT a classification for the length of each side of a small pizza? A whole B irrational C rational D integer Pizza Size Area (in 2 ) Small 100 Medium 200 Large Extra Large 576 Side length (in.) 1-41 Holt McDougal Algebra 1

22 Name Date Class 1-5 Challenge How Much Fencing is Needed? A farmer wishes to raise alpacas. The alpaca is related to the llama and is best known for its fleece. Alpacas can be prey for dogs, wolves and coyotes, so it is important to build a fence around any alpaca farm. The farmer currently owns a plot of land in the shape of a square. He knows that the land measures 532,900 square feet. To give the alpacas more room to graze, he buys an adjacent plot of land, which is in the shape of a right triangle. Both plots of land are shown below. The side labeled b ft has a length of 700 feet greater than the side labeled x ft. The farmer needs to determine the perimeter so he can build a fence. Complete the steps below to find how much fencing the farmer needs. 1. Find the value of x. 2. Find the value of a. 3. Find the value of b. The equation a 2 + b 2 = c 2 relates the lengths of the sides of a right triangle. 4. Find the value of a Find the value of b Find the value of a 2 + b 2. (This is c 2.) 7. Find the value of c. Round to the nearest whole number. 8. Find the perimeter of the land. 9. How much fencing does the farmer need? 1-40 Holt McDougal Algebra 1

23 Name Date Class 1-5 Review for Mastery Square Roots and Real Numbers The square root of a number is the positive factor that you would square to get that number. the square root of 9 is 3 because 3 squared is 9 9 = 3 because 3 2 = 3 3 = 9 A negative square root is the negative factor that you would square to get the number. the negative square root of 25 is 5 because 5 squared is = 5 because ( 5) 2 = ( 5)( 5) = 25 To evaluate a square root, think in reverse. Ask yourself, What number do I square? Find 36. Find ( 6) 2 = ( 6)( 6) = 36 Think: What negative factor do you square to get 36? 36 = Think about the numerator and denominator separately = 4 Think: What number do I square to get 4? 9 2 = 81 Think: What number do I square to get 81? = = = 2 9 Combine the numerator and denominator to form a positive factor. 1. Complete this table of squares Complete this table of square roots Find each square root Holt McDougal Algebra 1

24 Name Date Class 1-5 Review for Mastery Square Roots and Real Numbers continued This flowchart shows the subsets of the real numbers and how they are related. To identify the classifications of a real number, start at the top and work your way down. Write all of the classifications that apply to the real number 4. 4 can be shown on a number line. It is real. 4 can be written as Its decimal representation terminates: 4 = is an integer. 4 so it is rational. 1 4 is a negative integer. Stop. There are no more subsets in the chart below negative integers. 4: real number, rational number, terminating decimal, integer Write all classifications that apply to each real number _ _ _ 1-39 Holt McDougal Algebra 1

25 Name Date Class 1-5 Practice C Square Roots and Real Numbers Find each square root , A student is painting one wall of a room. A square window is located in the middle of the wall to be painted. The window measures 3 feet on one side. The area of the wall, not including the window, is 80 ft 2. Find the height of the wall to the nearest tenth. Write all the classifications that apply to each real number Holt McDougal Algebra 1

26 Name Date Class 1-5 Practice B Square Roots and Real Numbers Find each square root A contractor needs to cut a piece of glass to fit a square window. The area of the window is 12 ft 2. Find the length of the side of the window to the nearest tenth of a foot. 8. A piece of cloth must be cut to exactly cover a square table. The area of the table is 27 ft 2. Find the length of the side of the table to the nearest tenth of a foot. Write all the classifications that apply to each real number Holt McDougal Algebra 1

27 Name Date Class 1-5 Practice A Square Roots and Real Numbers Find each square root Max needs to paint a wall that is shaped like a square. He knows that the area of the wall is 75 ft 2. He needs to find the height of the wall. Find the height of the wall to the nearest tenth of a foot. 8. Paula has some bricks to make a square patio. She knows she has enough bricks to cover 50 square feet. She needs to find how long to make the patio. How long should Paula make the patio? Round your answer to the nearest tenth of a foot. Write all the classifications that apply to each real number: Natural, Whole, Integer, Rational, Terminating, Repeating, and Irrational _ _ Holt McDougal Algebra 1