Practice A For use with pages 8 13 Name the operation that would be performed first. 1. 18 2 5 1 12 2. 5 p 8 4 4 3. 27 4 (14 2 5) 4. 36 4 4 p 3 5. 8 1 10 4 5 2 3 6. 4 p (7 2 3) 2 7. 15 2 8 1 4 8. 13 2 7 1 2 9. 6 p 4 4 8 10. 7 p 4 2 3 11. 32 2 9 4 3 12. 9 1 3 p 4 13. 6(8 2 3) 14. 20 1 3 2 15. 5 2 2 8 p 2 16. 6x 1 3 when x 5 2 17. 4b 2 1 when b 5 5 18. 5 1 2m 2 when m 5 6 19. 3y 2 2 2 when y 5 1 20. 5 p 2a 3 when a 5 3 21. 4c 2 2 2c when c 5 5 22. 10 1 n 3 when n 5 0.5 23. 40 2 32 } r when r 5 4 24. x 2 4 3 2 12 when x 5 9 Was the expression evaluated correctly using the order of operations? If not, find and correct the error. 25. 8 1 3 2 2 p 5 5 8 1 3 2 10 5 11 2 10 5 1 26. (8 1 10) 4 6 2 3 5 18 4 6 2 3 5 18 4 3 5 6 27. Baseball In your last three baseball games, you scored 4 runs, 1 run, and 1 run, respectively. The average number of runs you scored for the three games is given by the expression } 4 1 1 1 1 3. Find the average number of runs per game. 28. Knitting You are buying yarn for a knitting project. One skein of yarn in a basic color costs $3.99 and one skein of yarn in a fashion color costs $5.99. The total cost of buying 5 skeins of basic yarn and 2 skeins of fashion yarn is given by the expression 5 p 3.99 1 2 p 5.99. Find the total cost of the yarn. 29. Desk Lamp You are buying a desk lamp for your study area. The lamp is $32 but is on sale for 75% of the original price. If you pay 5% sales tax, the expression 1.05(0.75 p 32) represents the total cost of the lamp. Find the total cost of the lamp. 30. Bowling A local bowling alley charges $4.50 to play one game and $4 to rent one pair of shoes. The expression 4.5g 1 4 represents the total cost for you to play g games. How much would it cost you to bowl 4 games? 19
Practice B For use with pages 8 13 1. 16 4 8 p 5 2. 7 2 2 24 4 3 3. 5 1 4 0.3 4. 18 4 6 1 4 p 3 5. 13 2 15 4 5 1 9 6. 2 } 3 p 3 2 2 5 7. 8(6 2 2) 1 4 8. 28 2 3(4 1 5) 9. p 5 2 6 4 3 10. (11 1 15) 4 13 11. 35 2 3 2 p 2 12. 4 } 5 (3 p 20) 2 17 13. 3x 4 2 5 when x 5 5 14. 8m 3 4 6 when m 5 3 15. 200 2 3y 2 when y 5 8 16. 5c 2 2 2c when c 5 9 17. 3 p 18t 2 when t 5 1 } 3 18. 42 } n 1 n when n 5 6 19. 7(x 1 5) when x 5 10 20. 5a } a 2 6 when a 5 8 21. 4d 2 } d 1 1 when d 5 3 22. Was the expression evaluated correctly using the order of operations? If not, find and correct the error. 80 2 1 } 3 (15) 2 5 80 2 5 2 5 80 2 25 5 55 23. Tournament During a bowling tournament, you bowled three games with scores of 110, 130, and 129, respectively. Your average bowling score is given 110 1 130 1 129 by }} 3. What is your average score? 24. Painting Three weeks ago, an art supply store started selling a paint kit for 75% of the original price. Now the kit is 15% off of the sale price. The expression 0.75x 2 0.15(0.75x) represents the current price of the paint kit where x is the kit s original price (in dollars). Find the current price of the kit if it originally cost $48. 25. Crown Molding You are decorating the perimeter of the ceiling of your living room with crown molding. The expression 2x 1 2y represents the total amount of molding you need where x is the width of the room (in feet) and y is the length of the room (in feet). Find the total amount of wood you need if the room is 11 feet wide and 10.5 feet long. 26. Core Sample Before a structure is built on a plot of land, it is 3 in. sometimes necessary to test the surface beneath the plot of land to determine its integrity. So, it may be necessary to take a core sample which is cylindrical in shape. Find the volume of the core sample shown by using the expression πr 2 h where r is the radius (in inches) and h is the height (in inches) of the cylinder. Use 3.14 for π. 36 in. 20
Practice C For use with pages 8 13 1. 32 4 8 p 5 2. 14 1 72 4 9 3. 6 p 9 2 33 4 11 4. 22 1 3 p 5 2 16 5. (4 3 1 6 2 ) 4 0.5 6. 10 1 5 3 4 25 2 7 7. 3 4 2 8 4 8 } 3 1 6 8. 5[(4 1 9) 2 3 2 ] 4 2 9. 14 4 F 3 } 4 (17 2 9) 2 G 10. 0.2 1 0.6(3 2 1 4 2 ) 11. 3[7(14 2 2 3 ) 1 5] 2 18 12. 1.75 1 2[17 2 (5 2 3) 2 ] 13. 6(m 2 4) when m 5 19 14. 9y 3 1 2 when y 5 4 15. 10x 2 3x 2 when x 5 2 16. 7x 2 1 2x when x 5 1 } 2 17. 6(m 2 1) } 2m 1 3 when m 5 6 18. 5t 2 1 2 } t 2 3 when t 5 2 19. 5x 2 2 3y 2 when x 5 8 and y 5 4 20. 5m 1 4 } 3n when m 5 4 and n 5 10 21. Was the expression evaluated correctly using the order of operations? If not, find and correct the error. 2 } 3 (9 2 2 2 3 ) 1 8 5 2(3 2 2 2 3 ) 1 8 5 2(9 2 8) 1 8 5 2(1) 1 8 5 10 22. Picture Frames You are purchasing wood to make 5 picture frames that are all the same size. The expression 5(2x 1 2y) represents the total amount of wood you need to make the frames where x is the width of a frame (in inches) and y is the length of a frame (in inches). Find the amount of wood you need to make the frames if each frame is 8.25 inches wide and 10.5 inches long. 23. Postal Service The United States Postal Service requires that the dimensions of a package mailed by parcel post using regular rates meet certain standards. One standard is that the expression h 1 (2l 1 2w), where h is the package s height, l is the package s length, and w is the package s width, cannot be greater than 108 inches. Does the package below meet this standard? 24.5 in. 40.5 in. 28 in. 24. Golfing One way a golfer can evaluate his or her playing is by using a number called a handicap. The first step to calculating your handicap is to calculate your handicap differential. The handicap differential is given by the expression [(s 2 c) p 113] 4 l, where s is your game score, c is the course rating of the course you played, and l is the slope rating of the course. You played a course with a 72.1 course rating and a 129 slope rating. Your score was 90. What is the handicap differential for the game? Round your answer to the nearest tenth. 21
Challenge Practice For use with pages 8 13 In Exercises 1 and 2, insert grouping symbols in the expression 4 1 7 3 3 2 2 2 1 5 3 6 so that the value of the expression is a. 1. a 5 25 2. a 5 83 3. Insert grouping symbols in the expression 24 4 3 1 9 2 4 3 6 1 10 so that the value of the expression is (a) the minimum value and (b) the maximum value. 4. A type of tree doubles its diameter every 10 years. If the tree is currently 1 inch in diameter, in how many years will the tree be 16 inches in diameter? 5. Due to improvements in design and manufacturing, the speed of a new computer doubles every 1 } 1 2 years. In how many years from now will computers be approxi mately 1000 times faster than they are now? 6. Due to improvements in manufacturing, the cost of a model of computer decreases by } 1 2 every 1 } 1 2 years. How many years will it take before a $2000 computer decreases in price to $250? 26
Investigating Algebra Activity: Evaluating Expressions For use before Lesson Materials: paper and pencil QUESTION How can you evaluate an expression? EXPLORE Evaluate expressions Evaluate each expression. (a) 12 2 5 3 3 1 10 (b) 24 4 8 2 2 3 9 (c) 2 3 6 2 5 3 1 (d) 6 3 12 4 3 2 16 1 4 STEP 1 Evaluate from left to right Evaluate each expression from left to right. (a) 12 2 5 3 3 1 10 5 7 3 3 1 10 5 21 1 10 5 31 (b) 24 4 8 2 2 3 9 5 3 2 2 3 9 5 1 3 9 5 9 (c) 2 3 6 2 5 3 1 5 12 2 5 3 1 5 7 3 1 5 7 (d) 6 3 12 4 3 2 16 1 4 5 72 4 3 2 16 1 4 5 24 2 16 1 4 5 8 1 4 5 12 STEP 2 Perform multiplication and division first Evaluate each expression from left to right, performing multiplication and division first and then going back and adding and subtracting. (a) 12 2 5 3 3 1 10 5 12 2 15 1 10 523 1 10 5 7 (b) 24 4 8 2 2 3 9 5 3 2 2 3 9 5 3 2 18 5215 (c) 2 3 6 2 5 3 1 5 12 2 5 5 7 (d) 6 3 12 4 3 2 16 1 4 5 72 4 3 2 16 1 4 5 24 2 16 1 4 5 8 1 4 5 12 STEP 3 Compare results As you can see, in parts (a) and (b) you get two different answers to the same problem while in parts (c) and (d) the answers are the same. Everyone needs to get the same answer, so the rule is to perform the operations in the same way it was done in Step 2. Use your observations to complete these exercises Evaluate the expression by first multiplying and dividing and then adding and subtracting. Then evaluate the expression from left to right and tell which expressions have incorrect answers. 1. 9 3 3 2 14 1 8 2. 42 4 7 2 2 3 4 3. 72 4 9 3 4 2 10 4. 5 3 8 1 48 4 4 5. 15 1 5 3 2 2 7 6. 36 2 12 4 3 1 21 17
Study Guide For use with pages 8213 GOAL Use the order of operations to evaluate expressions. Vocabulary The order of operations was established to evaluate an expression involving more than one operation. Order of Operations STEP 1 Evaluate expressions inside grouping symbols. STEP 2 Evaluate powers. STEP 3 Multiply and divide from left to right. STEP 4 Add and subtract from left to right. EXAMPLE 1 Evaluate expressions Evaluate the expression 4 2 p 5 2 6 2. Solution 4 2 p 5 2 6 2 5 16 p 5 2 36 Evaluate powers. 5 80 2 36 Multiply. 5 44 Subtract. EXAMPLE 2 Exercises for Example 1 1. 20 2 3 2 1 7 2. 5 p 2 3 4 6 3. 4 p 6 2 21 4 3 Evaluate expressions with grouping symbols a. 47 2 2(9 1 12) b. 6[2 3 1 (13 2 8)] Solution a. 47 2 2(9 1 12) 5 47 2 2(21) Add within parentheses. 5 47 2 42 Multiply. 5 5 Subtract. b. 6[2 3 1 (13 2 8)] 5 6[8 1 (13 2 8)] Evaluate power. 5 6[8 1 5] Subtract within the parentheses. 5 6[13] Add within the parentheses. 5 78 Multiply. 22
Study Guide continued For use with pages 8213 Exercises for Example 2 4. 3(14 2 5) 5. 6(9 2 1 4 ) 6. (7 1 5) 2 (8 1 4) 7. (3 3 2 6) 4 3 8. 4 2 (2 1 8) 9. 9[15 4 (2 1 3)] EXAMPLE 3 Evaluate an algebraic expression Evaluate the expression 4y 1 8 } 2 1 y when y 5 3. Solution 4y 1 8 } 5 } 4(3) 1 8 2 1 y 2 1 3 Substitute 3 for y. 5 12 1 8 } 2 1 3 Multiply. 5 20 } 5 Add. 5 4 Divide. Exercises for Example 3 Evaluate the expression when w 5 9. 10. 17 2 3w 11. w 2 2 13 12. 5w } w 1 6 13. 7(13 2 w) 14. 2w 2 2 15 15. 5w 2 1 } 3 w 23
Math and History Application For use with pages 8 13 History Few details of Muhammad ibn Musa al-khwarizmi s life are known. He was born around 790 A.D. in the town of Khwarizm (now Khiva), in the Khorasan province of Persia (now in Uzbekistan) and died around 840 A.D. His family moved to a place near Baghdad soon after he was born. He studied at the House of Wisdom in Baghdad and wrote on several subjects including mathematics, astronomy, cartography, and geography. One of his mathematics books, titled Al-Jebr W al-muqabala, gave us the word algebra from al-jebr. The word al-jebr originally meant to move a negative term to the other side of an equation. The word al-muqabala meant to balance an equation. In time however, the term al-jebr, or algebra, came to include all operations involved in modern algebra. The purpose of the book, in al-khwarizmi s own words, was to teach: what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of the lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned. The book was intended to be highly practical and algebra was introduced to solve real-life problems that were part of everyday life in the Arab empire at that time. How modern algebra is taught and the use of algorithms can be traced to al-khwarizmi s work. Math In Al-Jebr W al-muqabala, al-khwarizmi did not use any symbols, not even numerals, and he expressed everything in words. This book was used as an algebra text for hundreds of years. He used a systematic and logical approach to solving linear and quadratic equations which is still used today. 1. Explain why al-khwarizmi may have written an algebra book without using any symbols. If your algebra book was written without using symbols, estimate how many pages would be needed. 2. The steps taken to solve an expression such as } 3 p 8 2 4 2 5 10 may have been written in al-khwarizmi s Al-Jebr W al-muqabala as follows. I multiplied three and eight and got twenty-four, then subtracted four and got twenty; finally I divided twenty by two and got ten. Write two algebraic expressions and evaluate. Next, write the steps taken to evaluate the expression in words without using symbols. 25