Guide Link Learn o Multiply integers. Simplify algebraic expressions. ~ Reag-Wor~d Line Hot air balloon races grab the attention of people from miles around. During a balloon race, pilots must fly over a target on the ground and drop a marker as close to the target as possible. During a recen~ race, the winning oalloon descended at a rate of about 8 meters per secona. Core EE.3 Multiolication is reoeated addition. So. 3(-40) means that -40 is used as an addend 3 times. 3(-40] = -40) + -40) + (-40) = -120-20 -80-40 0 40 By the Commutative Property of Multiplication, 3/-40/= -40(3). This and other similar examples demonstrate the rule for multiplying integers with different signs. Words The product of two integers with different signs is negative. Examples 2(--6) = --12 --2(6) = --12 Find each product, a. -3.12 3(12) - -36 b. 4(-7) 4(-7) -- -28 GO~ t~ ~ Do these problems to find out. lb. -6-12 Multiplying Integers
s The product of two integers with the same sign is positive. Exampl~... e~s 4-6 = 24 --41--6) = 24... /) The oroduct of two positive integers is positive. What is the sign of the product of two negative integers? Look at the pattern below. Factor Rule: Times-5 Product 2-5(2) --10 0 5(01 _1-5(-1 5 --2 5(--21 10 Find each product. a. -5(-7) -5(--7) = 35 --8(--14) -8(--14)=112 6#{ ~{? Do these problems to find out. 2a. --5(--11) 2b. -13(-4) An airplane descends at a rate of 175 feet per minute. What is the airplane s change in altitude after 5 minutes? Write a multiplication expression. Then find and interpret the product. The word descends means downward, so the rate per minute is represented by a negative integer. Words 5 minutes times descent rate of!75 feet per minute Expression 5 x (-175) So, the airplane descends 875 feet after 5 minutes. 6e{ ~? Do this problem to find out. A scuba diver descends from the surface at a rate of 7 feet per minute. What was the scuba diver s depth after 15 minutes? Write a multiplication expression. Then find and interpret the product. Chapter 2 Operations with Integers
Use the Commutative and Associative Properties of Multiplication to multiply more than two integers. Find -4(12)(-5). Look for products that are multiples often to make the multiplication (~[~~--~.~ Use the Associative Property --4(12)(--5) = [-4(12)](--5) = --48(--5) = 240 Use the Commutative Property -4(12)(-5) = -4(-5)(12) = 20(12) = 240 ~0# I~ Do these problems to find out. 4a. --7(9)(--6) 4b. -3(-4)(-5) You can use the rules for multiplying integers to simplify and evaluate algebraic expressions. Simplify --Ta(4b). -7a(4b) = (-7)(a)(4)(6) = (-7.4)(a- b) : -28ab ~0# I~ Do these problems to find out. 5;b. - 9x(3y) Evaluate 3xy ifx = 2 and y = --6. = [3(2)](--6) < ~0~ /~ Do these problems to find out. 6a. Evaluate 2rs if r = 5 and s = - 10. 6b. Evaluate 4ab if a = -8 and b = -4. Lesson 2-4 Multiplying Integers
3. 8(-3)(-5) Mr. Heppner bought lunch with his debit card every day for 5 days. Each day he spent $8. If these were his only transactions, what was the change in his account balance? Write a multiplication expression.then find and interpret the product. /~x~m!~! ~:~) Simplify each expression. 0x,, rq~i~ 1 6. -2.9m 7. -30(7b) 8. -6e(-4f) 9. 8j, ifj=--11 10. -9cd, if c = -3 and d -- -7 Solutions Find each product. ~3(--9) 12. 8(--9) 13. 25.3 14. -4(--8) 15.--7(-7) 16. 2(-11)(5) 17. --8(--7)(--6) 18. -8(-20)(5) 19, 20. ~ T~" ~ ~ ~I The temperature dropped 2 F every hour for the last 6 hours. What is the total change n temperature?write a multiplication expression.then find and interpret the product, ii~, ~ ~ ~i An elevator takes passengers from the ground floor down to an underground parking garage. Where will the elevator be in relation to the ground floor after 5 seconds if it travels at a rate of 3 feet per second? Write a multiplication expression. Then find and interpret the product. Simplify each expression. ( ~i:~, } 21. 5(-6m) 22. -5 10s 25. 4~(b)(-9) 26. -12(-j)(-3k) Et, alue~e each expression. 2g. --9m(--9n) 27. 3e(--2f)(gg) 24. 1 Ia(7c) 28. 3r(7s)(5t) 29. 10n, ifn = --I0 30. 8m, if m = --5 31, 4xyifx= -6 andy = 3 32. --15st, ifs = 4and t = -9 33. 34. g~,~ ~dej ~[t~ ~a~ rsat[es Diego borrowed $780 from his father to buy a new saddle for his horse. He earns $65 each weekend cleaning stables and uses his earnings to pay off his debt. Write an expression Diego can use to determine how much he owes each week.then make a table to show how much he owes after 2, 4, 6, and 8 weeks. Based on the pattern in the table, predict how long it will take Diego to repay his father. The table shows the number of Calories burned per minute for a 120-pound person during different activities, What is the change in the number of Calories in a 120-pound person s body if he runs for 20 minutes and swims for 25 minutes? Interpret the product. Activity Calories per ~Jnute ballet dancin~ 6 bicycling 12 ~unning 18 swimming 8 Operations with integers
(-5) card every day for, what was the 0n,Then find and 14. -4(-8) 18. -8(-20)(5) Replace each with <, >, or = to make a true sentence. 3S. --4(6) ~, 3(8) 36. 3(--2) ;, 4(--I) 37. (--I I)(--3)(6)(--2) --24(18) :38. (--5)(--2)(9)(4)(10)(--9) ~ ~t,$~ ~eaeon Ab~t~ae~:l~ Two teams of students played a trivia game. Each team earned 5 points for every correct answer, lost Team Correct Incorrect ]Passed 8 points for every incorrect answer, and lost 2 points for every I 12 3 passed question. For each team, write an expression to determine the number of points they earned Which team won?,,~:: The price of stock fell $2 each day for 14 consecutive days. The original price of the stock was $41. Write an expression that you could use to find the price of the stock on any day. 41. ~gg~ ~u[t[p[e l~ep~e~e~at[o~8 In this problem, you will investigate the relationship between time and altitude. A hot air balloon is at an altitude of 600 feet It begins descending at a constant rate of 15 feet per minute. a. Table Complete the table below that shows the altitude every S minutes. Altitude (feet) 600 parking if it Words What does 0 minutes represent? c. Graph Graph the ordered pairs. d. Symbols Determine the time it will take for the balloon to reach the ground. Explain how you solved. 1 u(7c) 42. ~ ~demt~fy ~ract~e Name two integers that have a product between -10 and -15. 43, ~J~ ~en~:~fy g~et~e Name all of the values ofx if 71 x I = 63. 44. ~ ~ef~eve~e ~t~e I~ob~e~n~ Positive integersa and CsatisfyA(A - C) = 23. What is the value of C? 45. ~ ~demtify 9t~oet~e Calculate (-10)(5)(18)[7 ( 7)] mentally. Justify your answer. g~ &~ Ose a ~3o~terexa~e Determine whether each of ~he following is true or false. If false, give a ounterexampleo 46. The product of three negative integers is positive. 47. The product of four negative integers is positive. 48. ~ 8~Idi~ o~ l:~e E~e~a~ ~e~ie~ When multiplying more than two integers, how can you determine the sign of the product? Lesso. 2-4 Multiplying integers
50. A submarine descends at a constant rate of 300 feet per minute, Which of the following equations could be used to find the altitude of the submarine after 5 minutes? 5(--300) =--1500 5(300) = 1500 --5(300) = -- 1500-5(--300) = 1500 Which of the following equations is modeled by the number line below? 51. Simplify - 14(-2)(-12). A-336 C 168 B -168 D 288, The distance from the water line to the bottom of a cargo ship changes based on the weight of the cargo. weight0f Dept,0fShip! j ~rgo (t~s! i (feet) i -18-12-6 0 6!2 18 F -2(-6)=--12 H 2{6)=--12 --2(6)=-12 J 2(--6)=--12 If the pattern in the table continues, find the depth of the ship if the ship is carrying 100 tons of cargo. +::~... 53. The highest point in California is Mount Whitney, with an elevation of 14,494 feet. The lowest point is Death Valley, with an elevation of -282 feet. How much greater is the elevation of Mount Whitney than Death Valley? 7.NS.ld 54. 6+(-9)+9 56, 9+ 16+{-10) 55. -7+(-13)+4 57. --12 + 18 + (--12) 58. The table shows the lowest recorded temperatures for certain states. 6.NS.7b State Station Date Temperature ( F) Alaska Prospect Creek Camp Jan. 23,1971-80 Montana Rogers Pass Jan. 20,1954-70 I Wisconsin Danbury Jan. 24,1922-54 ao Compare the lowest temperature in Montana and the lowest temperature in Wisconsin using an inequality. Write the temperatures in order from greatest to least. Find each difference 7,NS.lc 59. -12-16 61. 8--10 63. Challenger Deep is 10,924 meters below sea level. Write an integer to represent this depth of Challenger Deep.Then identify its opposite and describe the meaning. 6.Ns.s