Arithmetic and Chapter, Lesson 6 + = false 0 5 = true n = open Write true if the statement is true or false if it is false. Write open if the statement is neither true nor false.. + 5 = 6 6. 50 5 = 45. + = 6 7. 8 = 6. 7 7 = 0 4. 6 6 = 6 5. + n = 7 6. 8 = 4 7. = 5 8. 0 + 0 = 60 9. n = 44 0. 7 9 =. 7 9 = 8. 6 6 = 8 8. =. + n = 5 9. 00 + 0 = 0 0. 7n = 49. 7 7 = 49. 7 n =. n = 6 4. 60 60 = 0 5. 7 = 8 4. 5 = 4 5. 8 9 = 6. 8n = 7. 4 4 = 8 8. 4 n = 9. 6 = 0. 7 + 7 = 5 AGS Publishing. Permission is granted to reproduce for classroom use only
Representing Numbers Using Letters Chapter, Lesson Numerical expressions: 6 ic expressions: m 5 8d + Variables: n in 7n + 7 k in k Operations: Multiplication and addition in y + Division in 4 Name the variable in each algebraic expression.. 4y +. k 6. x + 7 4. 7n 5. 4m 6. (d) r 7. 4 8. 4k 0 9. x 00 0. p. 4 + y. m 5 Fill in the table. For each expression, write the expression type numerical or algebraic and list the operation or operations. Expression Expression Type Operation(s) 6. 4. 8d 5. 6. 5 + 7. 8. 6 9. 0. p.. 4k + 4. 4. Solve the problem. 5. Only 7 members of Mr. Ricardo s class are going on the class trip. The class has a total of k students. Write an algebraic expression for the number of students who are not going on the trip. AGS Publishing. Permission is granted to reproduce for classroom use only
Integers on the Number Line Chapter, Lesson 8 7 6 5 4 0 4 5 6 7 8 All the numbers on this number line are examples of integers. An example of a negative integer is 5 (see arrow). An example of a positive integer is 5 (see arrow). The number 0 is neither negative nor positive. 5 = 5. In other words, 5 is 5 units from 0 (count the units). 5 = 5. In other words, 5 is 5 units from 0 (count the units). Identify each integer as either negative, positive, or zero.. 6.. 4. 5. 8 6. 9 7. 0 8. 9. 0 0.. 7. Write each absolute value.. 5 4. 6 5. 6. 7. +8 8. 5 9. 0... 4. 9 4. 9 Solve this problem. 5. On the number line, how could you represent $5 that you earned? How could you represent $5 that you had to pay? AGS Publishing. Permission is granted to reproduce for classroom use only
Adding Integers Chapter, Lesson 4 4 Add + ( 5). Start at, move 5 units to the left. The answer is. 5 8 7 6 5 4 0 4 5 6 7 Add + 7. Start at, move 7 units to the right. The answer is 4. +7 8 7 6 5 4 0 4 5 6 7 Answer the questions.. To add a negative number, in which direction do you count on the number line?. To add a positive number, in which direction do you count on the number line? Write each sum on the blank.. 4 + 4 4. + ( 7) 5. + 5 6. 0 + 6 7. + ( 5) 8. 5 + ( ) 9. 5 + 0. 6 +. + ( ). + ( 4). 6 + 6 4. 6 + ( 6) 5. 4 + 8 6. 6 + 7. + 6 8. + 9 9. 7 + 7 0. + ( 8) AGS Publishing. Permission is granted to reproduce for classroom use only
Subtracting Integers Chapter, Lesson 5 5 Find the difference: 4 ( 5) Rule To subtract in algebra, add the opposite. 5 is the opposite of 5. 4 + 5 = 9 Rewrite each expression as addition. Solve the new expression.. 4 ( ) 0. 5 ( 5). 9 (+). 4. 6 (+0) 5. 7 ( 0) 6. 4 (+4) 7. (+8) 8. ( ) 9. 6 (+). (+9). (+4). 6 8 4. 8 ( ) 5. (+7) 6. 8 ( 7) 7. 0 (+5) 8. 5 6 Solve these problems. Write an expression and the answer. 9. Dara s kite is flying 67 feet high. Jill s is flying 40 feet high. What is the difference between the heights of these two kites? 0. A helicopter hovers 60 m above the ocean s surface. A submarine is resting 0 m underwater, directly below the helicopter. What is the difference between the positions of these two objects? AGS Publishing. Permission is granted to reproduce for classroom use only
Multiplying Integers Chapter, Lesson 6 6 Notice the possible combinations for multiplying positive and negative integers. positive (positive) = positive 4(4) = 6 positive (negative) = negative 4( 4) = 6 negative (positive) = negative 4(4) = 6 negative (negative) = positive 4( 4) = 6 Multiplying any integer, positive or negative, by 0 gives 0 as the product. Tell whether the product is positive, negative,or zero.. (4)( 7) _. ( 6)() _. ( 7)(0) _ 4. ( 9)( 9) _ 5. ()( ) _ 6. (5)( ) _ 7. ( 5)( ) _ 8. (0)(4) _ 9. ( 5)(7) _ 0. ( 8)( ) _. ( )( 9) _. (4)(6) _. ()( ) _ 4. ()(9) _ 5. ( 6)(4) _ Find each product. Write the answer. 6. ()(9) 7. ( 5)(9) 8. ()( 9) 9. ( 0)() 0. ( 9)( 7). ( 5)(0) 5. ( )( ) 6. ( 4)(8) 7. ( 0)( 8) 8. (5)( 5) 9. ( )( ) 0. (7)(0). ()(). ( )(0) 4. (6)( 4) AGS Publishing. Permission is granted to reproduce for classroom use only
Integers Chapter, Lesson 6 7 Data from a climbing expedition is shown in this table. Elevation in Feet (Compared to Sea Level) Base Camp 84 Camp +5,07 Camp +7,5 Camp +8,860 Camp 4 +0,0 Camp 5 +0,856 Camp 6 +,49 Summit +,05 During the climb, some climbers began at base camp and climbed to the summit. Other climbers also began at base camp but did not reach the summit these climbers moved back and forth between camps carrying supplies and other necessities. The movements of various climbers in the expedition are shown below. Find the number of feet climbed by each climber.. Climber A: Base Camp to Camp to Base Camp. Climber C: Base Camp to Camp 5 to Base Camp. Climber F: Base Camp to Camp to Base Camp 4. Climber B: Base Camp to Camp to Base Camp 5. Climber H: Base Camp to Camp 6 to Base Camp 6. Climber E: Base Camp to Summit to Base Camp 7. Climber G: Base Camp to Camp 4 to Base Camp 8. Climber D: Base Camp to Camp 5 to Camp to Camp 4 to Base Camp 9. Climber I: Base Camp to Camp to Camp to Camp 6 to Base Camp 0. How many feet above base camp is the summit? AGS Publishing. Permission is granted to reproduce for classroom use only
Dividing Positive and Negative Integers Chapter, Lesson 7 8 Notice the possible combinations for dividing positive and negative integers. positive positive = positive 6 = positive negative = negative 6 = negative positive = negative 6 = negative negative = positive 6 = Dividing 0 by any integer, positive or negative, produces 0 as the quotient. Tell whether the quotient is positive, negative,or zero.. 6 4. 6 9. 0 4. 5. 4 6. 00 0 7. 6 9 9. 7 0. 0 4. 8 9. 9. 56 8 4. 500 5 5. 8 8. 5 5 Find and write each quotient. 6. 6 7. 7 8. 8 9. 5 7 0. 4. 6 8. 45 9 4. 50 0 5. 7 9 6. 4 7. 0 6 8. 7 9 9. 0. 9. 00 00 AGS Publishing. Permission is granted to reproduce for classroom use only
Simplifying Expressions One Variable Chapter, Lesson 8 9 Simplify n + + 4n.. Look for like terms. n and 4n are like terms, because they have the same variable, n.. Combine the terms: n + 4n = 6n. Rewrite the whole expression: 6n + Now you are finished, because 6n cannot combine with. In each expression, underline the like terms.. k 8 + k. p + + p. 00 + 4w + 4w 4. 5m + m 5. 7x + 5x 6. + c + c 7. 4 + m + m 7 8. y ( y) + 7 9. 4x + 5x 0. 8r + ( r) Simplify each expression.. b + b. y + y + y. 7j + j j 4. k 7 + k 5. x + x 4 6. + d + 8d 7. 9g + ( g) + 4 8. 4h h 9. m + ( 8m) 0. + 4k k. y + ( y) + 5. 5 + 6n 4n. x + x 4. h + 7h 5. ( k) + k 6. 7d d + 40 7. 8 + m ( m) 8. w + ( 5w) 9. + 5x x 0. 8g + ( 5g) 6 AGS Publishing. Permission is granted to reproduce for classroom use only
Simplifying Expressions Several Variables Chapter, Lesson 9 0 Simplify j + 4 + j + k.. Scan for variables. The expression has two: j and k.. Combine j terms: j + j = j. Combine k terms: k (no combining required) 4. Combine integers: 4 + ( ) = + 5. Rewrite the whole expression: j + k + Now you are finished, because you cannot combine unlike terms. Check the column or columns to show which kinds of terms each expression includes. Expression x terms y terms Integers. x + x + + 6y. y 4. x + y 0 + y 4. 7 68 5. x 8 Combine like terms. Simplify each expression. 6. k k + + 5r 7. c + b + 8b + c 8. 5k + + j k 9. 7 + 4p 6 + m p 0. 4n + 4 ( d) + n 5. 8 + ( 4x) + x + 8y. w + ( ) + 8y + w. 6h 9h + 8k + ( k) 4. m m 8p p 5. 4x 7 + y 7x 7x AGS Publishing. Permission is granted to reproduce for classroom use only
Positive Exponents Chapter, Lesson 0 To multiply like variables having exponents, add the exponents. k k = k ( + ) = k 6 To divide like variables having exponents, subtract the exponents. m 4 m = m (4 ) = m You cannot use these rules to multiply or divide unlike variables. y a b 5 j Is the rewritten expression on the right true or false? Write the answer.. m 4 m = m (4 + ). a a a = a ( + + ). k n = k ( + ) 4. y 7 y = y (7 ) 5. b 8 b = b (8 ) 6. a a = a 7. d 6 d = d (6 + ) 8. w w 4 w 5 = w ( + 4 + 5) Simplify each expression. 9. k k 0. w 5 w. j 4 j 0. n 6 n. d d 5 d 7 4. y 7 y 5. x x x 8 6. a a a 4 a 4 Use a calculator to find the value of each expression. 7. x when x = 7 8. c 5 when c = 9. r 5 when r = 0. k when k = 0.8 AGS Publishing. Permission is granted to reproduce for classroom use only
Formulas with Variables Chapter, Lesson Find the perimeter of a triangle with unequal sides. The perimeter is the sum of the length of all the sides. a c Perimeter = a + b + c b If a =, b = 5, and c = 4, then the perimeter of the triangle is + 5 + 4 =. Use the formula for the perimeter of a triangle with unequal sides to find each answer.. a = 8 cm b = 0 cm Perimeter = c = 6 cm. a = m b = 7 m Perimeter = c = m Use the formula for the perimeter of a square. Write each answer. s. s = 5 km s s Perimeter = s Perimeter of a square = 4s 4. s = 8 cm Perimeter = 5. s = 9 m Perimeter = AGS Publishing. Permission is granted to reproduce for classroom use only
Commutative Property of Addition Chapter, Lesson n + n = Expanded notation: n + n + n + n + n = n + n + n + n + n n + n = 5n Commutative property of addition: 7a + 6a = 6a + 7a Find each sum using expanded notation.. k + k _. r + r _. 4y + y _ 4. n + 5n _ Rewrite each sum showing the commutative property of addition. 5. b + 7 6. k + 5 7. x + 5x 8..7 +. 9. 5m + p 0. 7q + 6. 64 +. 8y + 4y Solve these problems.. Torri weighs 98 pounds. Carrie weighs 6 pounds. Torri adds her weight to Carrie s weight. What sum will she get? 4. Suppose Carrie adds her weight to Torri s weight. What sum will she get? 5. What mathematical property do the sums in problems and 4 illustrate? AGS Publishing. Permission is granted to reproduce for classroom use only
Commutative Property of Multiplication Chapter, Lesson 4 Turning a rectangle two different ways shows that the order of factors in a product can change without affecting the result. In each case, the product is the area of the rectangle in square units. Factors Product Factors Product 5 = 5 5 5 = 5 5 For each pair of rectangles, write two products representing area in square units.. = sq. units 7 = sq. units 7. 6 = sq. units 6 = sq. units Answer the questions. Mrs. Rossi buys 6 bags of apples. Each bag holds 8 apples. Mr. Lundgren buys 8 bags of apples. Each bag holds 6 apples.. How many apples did Mrs. Rossi buy? 4. How many apples did Mr. Lundgren buy? 5. What mathematical property do problems and 4 illustrate? AGS Publishing. Permission is granted to reproduce for classroom use only
Associative Property of Addition Chapter, Lesson 5 ( + 0) + 4 = + (0 + 4) (b + c) + d = b + (c + d) Rewrite each expression to show the associative property of addition.. k + (k + 5). (. + 8.) + 6.6. (x + 0y) + 4 4. 7 + (5 + q) 5. ( + n) + 6. g + ( + h) Answer the questions. In a club, members bring all of the sandwiches for a picnic. Mike and Lynn arrive together. They have already put together Mike s sandwiches and Lynn s 5 sandwiches. Hosea comes a little later with 4 sandwiches. In all, the club has a total of sandwiches. 7. Write an addition expression that shows the grouping described above. 8. Suppose that Mike had come first, alone, and that Lynn and Hosea had come with their combined sandwiches later. Write an addition expression to represent this grouping. 9. Would the club s total number of sandwiches be the same with either grouping? 0. What mathematical property does this story illustrate? AGS Publishing. Permission is granted to reproduce for classroom use only
}6 Seats } Name Date Period Workbook Activity Associative Property of Multiplication Chapter, Lesson 4 6 (5 ) = 5( ) (pq)r = p(qr) Use the associative property of multiplication to place parentheses in two different ways.. mnq. abc. 4st 4. kvz 5..5bc 6. 9mn 7. aks Answer the questions to solve the problem. A college will build a theater with at least 0 balcony seats. Two architects have drawn up plans. Here are their designs for the balconies: Plan A: rows Plan B: 6 Seats rows 8. Which plan does ( 6) describe? Hint: Think of the parentheses as inside dimensions. 9. Which plan does (6 ) describe? 0. Do the two plans have the same number of balcony seats? AGS Publishing. Permission is granted to reproduce for classroom use only
The Distributive Property Multiplication Chapter, Lesson 5 7 4( + 8) = 4 + 4 8 = 8 + = 40 5(x + y) = 5 x + 5 y = 5x + 5y Fill in the blanks in each rewritten expression.. (x + y) = x + y. 4(m + 4) = 4 + 6. 7(8 + ) = 56 + 4. 5(4 + c) = 0 + 5. (p + q) = q 6. 8(a + ) = 8a + 7. ( j + k) = k 8. 0(40 + 0) = 400 9. [ q + ( )] = 6 0. 4( + n) = 4n. ( 7 + b) = +. 7(d + k) = 7d +. 4(5 + ) = 4 4. (n + p) = p 5. k( + m) = k + 6. b(y + z) = + bz Rewrite each expression, using the distributive property. Simplify where possible. 7. 6( + ) 8. 6(r + z) 9. (d + k) 0. ( + w). ( 4 + m). 8[ a + ( )]. 9(x + y) 4. 7(g + 0) 5. 8( v + 8) AGS Publishing. Permission is granted to reproduce for classroom use only
The Distributive Property Factoring Chapter, Lesson 6 8 4j + 4k = 4( j + k) x y = (x y ) Identify the common factor in each expression.. k + w. jn + kn. q + ( r) 4. b b 5. p + 46p 6. dx dy 7..9a +.9b 8. r.5r 9. 7m + 7v Draw a line to match the expression on the left with its factored form on the right. 0. 7x 7y a. a(x + y). ax + ay b. (x y). 4x + 4y c. (ax + y). x + y d. 7(x y) 4. ax + y e. 4(x + y) Solve the problem. 5. Two children are paid their allowances in dimes and nickels. Each child receives exactly the same number of each type of coin. Let d stand for the number of dimes each child received. Let n stand for the number of nickels each received. One way to represent total amount of allowance to the children is d + n. What is another way? (Hint: Use the distributive property to factor.) AGS Publishing. Permission is granted to reproduce for classroom use only
Properties of Zero Chapter, Lesson 7 9 Additive Property of Zero: 4 + 0 = 4 + 0 = Additive Inverse Property: + = 0 5 + ( 5) = 0 Multiplication Property of Zero: 0(6) = 0 4(0) = 0 If the two numbers are additive inverses, write true. Otherwise, write false.. _. 7 7 _. 5 0 _ 4. 75 75 _ 5. 8 8 _ 6. x x _ Write each sum. 7. 9 + 0 _ 8. 0 + 7 _ 9. 0 4 _ 0. k + 0 _. 6 + 0 _. 0 + m _. 0 7 _ 4. k 5 + 0 _ 5. 0 y _ Write each product. 6. 0() _ 7. ( 7)(0) _ 8. 0 q _ 9. (xy)(0) _ 0. (0)( 9) _. (cde)(0) _. ( jk)(0) _. n 7 0 _ 4. (0)(ab ) _ Solve the problem. 5. Jenna said to Brett, I ll give you double the number of marbles you have in your pocket. Brett replied, But I don t have any marbles in my pocket. Jenna responded, So I ll give you double nothing, which is nothing. How could Jenna say the same thing in a mathematical expression? Underline one. a. + = b. 0 + = c. (0) = 0 AGS Publishing. Permission is granted to reproduce for classroom use only
Properties of Chapter, Lesson 8 0 Multiplication Property of : ()(7) = 7 ()(y) = y Multiplicative Reciprocals: 8, 8: 8 8 = x, x: x x = Complete the table by writing the reciprocal of the term and then checking your answer. Term Reciprocal Check. 4. 9. 7 4. 5. k 6. m 7. c 8. Solve the problems. 9. Each wedge of apple pie is of the pie. How many wedges make one 5 whole pie? Complete the equation to show your answer. = 5 0. In a geometry study group, 6 students were each given an identical puzzle piece of a hexagon (6-sided figure). The students assembled their pieces to make a whole hexagon. What fraction of the hexagon was each puzzle piece? Complete the equation to show your answer. 6 = AGS Publishing. Permission is granted to reproduce for classroom use only
Powers and Roots Chapter, Lesson 9 = ()() = 9 9 = = ()()() = 7 7= Fill in the blank in each sentence.. (9)(9)(9) = 6,859, so = 6,859.. If 7 7 = 89, then 7 = 89 and 89 =.. If 6 = 6, then 6 = 6. 4. If = 6, then the fourth of 6 is. 5. (4)(4)(4)(4)(4) =,04, so 4 =,04. 6. =, which means that =. Find each square root. You may use a calculator. 7. 49. 79 5. 0.4 8. 8. 4 6. 00 9. 6. 5,99 7. 8.5 0. 5 4..6 8. 6 Solve the problems. 9. Talia is sewing a quilt with a regular checkerboard pattern that is, all the squares are identical. In each square of the checkerboard, she plans to stitch a simple flower. Talia will have to stitch 6 flowers in all. How many squares lie along one side of the quilt? 0. The volume of a cube of sugar is.97 cm. Circle the letter of the expression that gives the length of one edge of the cube. a..97 b. (.) c..97 AGS Publishing. Permission is granted to reproduce for classroom use only
More on Powers and Roots Chapter, Lesson 0 Simplify the expression: ( x) Step ( x) = ( x)( x)( x) Step Multiply ( ) three times: ( )( )( ) = 8 Step Multiply x three times: (x)(x)(x) = x Step 4 Multiply the expanded number and variable: 8x Note: x = x or x x= x x = x Simplify each term. You can use a calculator.. (8d). ( 0n). ( y) 4. (m) 4 Find each value. Write all the possible roots. 5. 6 8. 8. 7 6. 9 9. k. 6 7. 8 0. 00. 6 Answer the questions to solve the problem. If a scientist built a machine that could transport people backward in time, then normal time might be represented as a positive number and backward time as a negative number. Suppose you could square or cube backward time. 4. What would be the square of 0 units of backward time? 5. What would be the cube of 0 units of backward time? AGS Publishing. Permission is granted to reproduce for classroom use only
Order of Operations Chapter, Lesson + 8 = Step Calculate the cube, or third power, of : = 8 Step Multiply: 8 = 96 Step Add: 8 + 96 = 04 Find the value using the order of operations... + 8 7. ( + 8)7 4. 64 6 4 5. + 6. + 7 9 7. 8 + 00 8. + 5 4 9. + + 4 0. (7 7 4) 5. (6 + 6) (59 55). 8 7 Answer the questions to solve the problem. Mr. and Mrs. Wang plan to knock out a wall between two rooms of their house to make one larger room. One room is a rectangle 0 feet by 5 feet, so its area in square feet is 0(5). The other room is a square, 9 feet on a side, so its area is 9 square feet. What will be the total area of the new room? 0 9 9 5. Circle the letter of the expression that calculates the answer. a. 0 + 5 9 b. (0 + 5 + 9) c. 0 5 + 9 4. State the order to perform the operations when calculating the answer. () () () 5. Work out the answer. (You can use a calculator.) square feet AGS Publishing. Permission is granted to reproduce for classroom use only
Order of Operations Chapter, Lesson 4 The answer shown below for the computation 0 + 6 is not correct. 0 + 6 6 8 The computation is not correct because the addition 0 + 6 was performed first. The order of operations states that the division 6 should have been performed first. The answer is when the computation is performed correctly. 0 + 6 0 + In each problem below, the computation has been performed incorrectly. For each problem, tell why the computation is incorrect. Then give the correct answer.. + 4 5 = 0. 0 5 + 5 =. 50 = 0 4. 4 + 6 = 0 5. 00 50 = 6. 5 8 + = 45 7. 00 00 + 50 = 75 8. 4 6 7 = 9. + 6 4 + 4 = 4 0. 0 4 + = 0 AGS Publishing. Permission is granted to reproduce for classroom use only
Writing Equations Chapter, Lesson 5 0 times some number equals 0. 0x = 0 Write an equation for each statement. Let x be the variable in the equation.. 6 times some number equals 0.. times some number plus 5 equals 9.. times some number minus 8 equals. 4. 7 subtracted from some number equals 4. 5. 0 times some number plus 7 equals 87. 6. subtracted from some number equals. x = 8 x = 4, 5, 6 ()(4) = 8 F ()(5) = 8 F ()(6) = 8 T 7. 5p = 5 Find the root of each equation by performing the operation on each value for the variable. Write T if the equation is true or F if the equation is false. 9. 0k = 80 p = k = 7 p = k = 8 p = k = 9 8. 7w = 49 0. a = w = 5 a = w = 6 a = w = 7 a = AGS Publishing. Permission is granted to reproduce for classroom use only
Solving Equations: x b = c Chapter, Lesson 6 Solve m = 8 for m. Step Write the equation. m = 8 Step Add to both sides of the equation. m + = 8 + Step Simplify. m = 0 Step 4 Check. 0 = 8 Solve each equation. Check your answer.. x 4 = 0. b 7 =. n = 7 4. k = 5. d 00 = 00 6. c = 0 7. y 4 = 4 8. r 80 = 0 9. w 8 = 0 Read the problem and follow the directions. 0. A sports store buys a shipment of catcher s mitts at the beginning of the year. By year s end, the store has sold 00 mitts and has 50 left. How many mitts did the store have at the first of the year? Let x stand for the number of catcher s mitts the store had at the beginning of the year: x 00 = 50 How would you solve this equation? Circle the answer. a. Subtract 00 from both sides. b. Add 00 to both sides. c. Subtract 50 from both sides. AGS Publishing. Permission is granted to reproduce for classroom use only
Solving Equations: x + b = c Chapter, Lesson 7 Solve k + 5 = 9 for k. Step Write the equation. k + 5 = 9 Step Subtract 5 from both sides of the equation. k + 5 5 = 9 5 Step Simplify. k = 4 Step 4 Check. 4 + 5 = 9 Solve each equation. Check your answer.. w + = 4. r + 8 =. y + = 7 4. c +.5 = 4.5 5. k + = 60 6. d + 9 = 8 7. n + 5 = 40 8. b + 7 = 4 9. x + = 8 Read the problem and follow the directions. 0. Amy read 7 books over the summer, more than Tim. How many books did Tim read? Let r stand for the number of books Tim read over the summer: r + = 7 How would you solve this equation? Circle the answer. a. Add to both sides. b. Subtract from both sides. c. Subtract 7 from both sides. AGS Publishing. Permission is granted to reproduce for classroom use only
Solving Multiplication Equations Chapter, Lesson 4 8 Solve 4y = 6 for y. Step Write the equation. 4y = 6 Step Multiply both sides of the equation by, the reciprocal of 4. 4 ( 4 )4y = 6( 4 ) Step Simplify. y = 4 Step 4 Check. 4 4 = 6 Note: Another way to do Step is to divide both sides of the equation by 4. 4 y = 6 4 4 Solve each equation. Check your answer.. q = 4. 0k = 0. 7x = 8 4. d = 8 5. 6w = 66 6. 4j = 6 7. 8y = 48 8. 5n = 0 9..p =.8 Read the problem and follow the directions. 0. Juanita is 5, five times the age of her brother, Frank. How old is Frank? Let a stand for Frank s age: 5a = 5 How would you solve this equation? Circle the answer. a. Subtract 5 from both sides of the equation. b. Multiply both sides of the equation by 5. c. Multiply both sides of the equation by 5. AGS Publishing. Permission is granted to reproduce for classroom use only
Solving Equations with Fractions Chapter, Lesson 5 9 Solve y = 5 for y. Step Write the equation. y = 5 Step Multiply both sides of the equation by the reciprocal of the fraction. ( ) y = ( )5 Step Simplify. y = 5 Step 4 Check. (5) = 5 Solve each equation. Check your answer.. x = 4. w = 8. q = 7 4. r = 6 5 5. 4 m = 0 5 6. 7 k = 4 8 7. d = 0 8. 6 y = 9. a = 4 5 Read the problem and follow the directions. 0. Spruceville received 5 inches of snow last winter, or 5 of its average 8 annual snowfall. What is Spruceville s average annual snowfall? Let n stand for Spruceville s average annual snowfall: 5 n = 5 8 How would you solve this equation? Circle the answer. a. Subtract 5 and then multiply by 5 8. b. Multiply both sides by 8 5. c. Subtract 5 from both sides. 8 AGS Publishing. Permission is granted to reproduce for classroom use only
Solving Equations More Than One Step Chapter, Lesson 6 0 Solve m 7 = for m. Step Add 7 to both sides. m 7 + (7) = + (7) Simplify: m = 8 Step Divide both sides by. m = 8 Simplify: m = 4 Step Check. (4) 7 = Simplify: = Solve each equation. Check your answer.. 4b + = 7. c + 8 = 4. 7x = 5 4. p + 8 = 8 5. 8v = 9 6. w 6 = 0 7. 5g + 5 = 5 One step is missing in the solution to each equation. In a complete sentence, write the missing step. 8. 7y 4 = 0 Step Step Add 4 to both sides of the equation. 9. d + = 9 Step Step Divide both sides of the equation by. 0. k + = Step Step Subtract from both sides of the equation. AGS Publishing. Permission is granted to reproduce for classroom use only
Equations Without Numbers Chapter, Lesson 7 ax b = c Solve for x. Step Write the equation. x 5 = 7 ax b = c Step Add 5 or b to both sides. x 5 + 5 = 7 + 5 ax b + b = c + b x = ax = b + c Step Divide each side by or a. x = a x = b + c a a Step 4 Check. x = 6 x = b + c a (6) 5 = 7 a( b + c ) b = c a 7 = 7 (b + c) b = c c = c Solve each equation for x. Check your answer.. ax c = b. bc = ax. x b + a = c 4. abx = c Follow the directions to solve the problem. 5. Center School has won two more soccer games than the combined wins of River School and Bluff School. This statement can be turned into a mathematical equation. Let x stand for the number of games Center School has won. Let y stand for the number of games River School has won. Let z stand for the number of games Bluff School has won. x = y + z + Solve the equation for z to show the number of soccer games Bluff School has won. AGS Publishing. Permission is granted to reproduce for classroom use only
Formulas Chapter, Lesson 8 Solve the formula for the perimeter of a triangle for c. P = a + b + c a + b + c = P (a + b) + c (a + b) = P (a + b) c = P (a + b) Solve each equation.. P = 6s for s. A = h(m + n) for m. P = b + h for b 4. A = lw for l 5. P = a + b + c for b 6. A = h(m + n) for n 7. V = hm for m 8. P = b + h for h Answer the questions to solve the problem. Mr. Jiang is building a deck on his house. He can calculate the perimeter of the deck using this formula: P = 4s 9. Solve the equation to show the length of a side that is, solve for s. 0. Is Mr. Jiang s deck in the shape of a square or a rectangle? Explain your answer. AGS Publishing. Permission is granted to reproduce for classroom use only
The Pythagorean Theorem Chapter, Lesson 9 Use the Pythagorean theorem: c = a + b Find c when a = 4 and b = 5. Use a calculator, and round the answer to the nearest tenth. c = (4) + (5) c = 6 + 5 c = 4 c = 4 = 6.404 Round off: 6.4 Use the Pythagorean theorem and a calculator to find the missing side of each triangle. Round to the nearest tenth.. a = b = 7 c =. a = b = 6 c = 0. a = b = 8 c = 4 4. a = 9 b = c = 6 Solve the problem. 5. A sailboat has a sail in the shape of a right triangle. You know that side a is m long and side b is 4 m long. How long is side c of the sail? Substitute known values in the Pythagorean theorem and solve. Use your calculator and round to the nearest tenth. c b a AGS Publishing. Permission is granted to reproduce for classroom use only
Using the Pythagorean Theorem Chapter, Lesson 9 4 The lengths of the sides of a triangle are ft, 4 ft, and 5 ft. Is the triangle a right triangle? When the lengths of the sides of a right triangle are given, the longest length is the hypotenuse. Substitute, 4, and 5 into the formula a + b = c. a + b = c + 4 = 5 9 + 6 = 5 5 = 5 True When the lengths of the sides of a triangle are ft, 4 ft, and 5 ft, the triangle is a right triangle. The lengths of the sides of a triangle are 0 cm, cm, and 5 cm. Is the triangle a right triangle? When the lengths of the sides of a right triangle are given, the longest length is the hypotenuse. Substitute 0,, and 5 into the formula a + b = c. a + b = c 0 + = 5 00 + 69 = 5 69 = 5 False When the lengths of the sides of a triangle are 0 cm, cm, and 5 cm, the triangle is not a right triangle. The lengths of the sides of various triangles are given below. Is the triangle a right triangle?. 4 in., 5 in., 7 in.. 5 cm, cm, cm. mm, 4 mm, mm 4. yd, yd, yd 5. 5 m, 68 m, 85 m AGS Publishing. Permission is granted to reproduce for classroom use only
Inequalities on the Number Line Chapter, Lesson 0 5 Write a statement of inequality for the number line. Use x as the variable. x 5 4 0 4 5 6 7 8 9 0. Write a statement of inequality for each number line. Use x as the variable. o 5 4 0 4 5 6 7 8 9 0. 5 4 0 4 5 6 7 8 9 0. o o 5 4 0 4 5 6 7 8 9 0 4. 5 4 0 4 5 6 7 8 9 0 Write T if the disjunction is True or F if it is False. 5. 5 6. 8 7. 4 8. 9. 0 5 0. 4 AGS Publishing. Permission is granted to reproduce for classroom use only
Solving Inequalities with One Variable Chapter, Lesson 6 Solve x 5 > for x. Step Write the inequality. x 5 > Step Add 5 to both sides of the inequality. x 5 + 5 > + 5 Step Simplify. x > 8 Note: For inequalities with addition, multiplication, or fractions, solve in the same way as for equations with the same operations. Solve each inequality.. x > 0. 5d > 0. k + < 4. 4q > 48 5. c + 40 6. g < 6 7. 7p < 8. w + 9 Solve the problems. 9. A school has arranged teaching loads so that no teacher ever has more than 5 students. Describe the school s teaching load using an inequality and the variable t. 0. The sponsor of a concert promises the concert singer a fee based on $5 per person in the audience. If attendance is below 00, however, the singer will be paid a minimum fee based on 00 seats filled. Using the variable f, write an inequality to represent the singer s minimum fee. AGS Publishing. Permission is granted to reproduce for classroom use only
Writing Equations Odd and Even Integers Chapter 4, Lesson 7 Two times a number added to 4 is 4. What is the number? Step Let n = the number. Then n is two times the number. Step Write and solve the equation. 4 + n = 4 4 4 + n = 4 4 n = 0 n = 5 Step Check: 4 + (5) = 4 4 = 4 Write an equation for each statement. Use n as the variable.. Three times a number added to is 5.. Four times a number decreased by 5 is 5.. Five added to 8 times a number is 5. 4. Seven times a number minus is 40. 5. Eleven times a number added to 0 is. 6. Eight times a number decreased by 5 is. 7. Four added to 9 times a number is 76. 8. Ten times a number decreased by is 99. 9. Nine times a number minus 4 is. Solve the problem. 0. A cook in a cafeteria has only 7 slices of rye bread left at closing time. An assistant immediately goes to a store and buys 4 identical loaves of sliced rye bread. With this additional supply, the cafeteria now has 79 slices of rye bread. How many slices are in one packaged loaf of rye bread? AGS Publishing. Permission is granted to reproduce for classroom use only
Using the % Solution to Solve Problems Chapter 4, Lesson 8 A train has traveled 50 miles toward its destination. This distance represents 0% of the total trip. What will the total mileage be? Step 0% of mileage = 50 Given. Step 0% 0 = 50 0 Divide both sides by 0 to solve for %. % of mileage = 5 Step 00% of mileage = 500 Multiply both sides by 00 to find the total mileage. Use the % method to find a number when a given percentage of the number is known.. 5% of a number is 00.. % of a number is.. 5% of a number is 5. 4. 98% of a number is 94. 5. 7% of a number is 5. 6. 44% of a number is 88. 7. 69% of a number is 69. 8. 0% of a number is 90. Solve the problems. 9. At Mayville Community College, 490 students are enrolled in the computer program. If 70% of the students in the college are in the computer program, how many students does Mayville Community College have in all? 0. An airline reports that 9% of its flying customers last year were under years of age. If 70,000 children under years of age flew on the airline s planes, how many customers did the airline have last year? AGS Publishing. Permission is granted to reproduce for classroom use only
Using the Percent Equation Chapter 4, Lesson 9 0% of the pencils in a box of 0 pencils are red. How many red pencils are there? Step Write the percent equation. p ( 00 )(n) = r Step Change the percent into a fraction. p = 0 0 ( )(n) = r 0 Step Write the total number. n = 0 ( )(0) = r Step 4 Simplify the fraction. ( 0 )(0) = r Step 5 Solve the equation. ( 0 )(0) = 6 00 00 0% of 0 is 6. There are 6 red pencils in the box. Use the percent equation to find the percent of each given number.. 0% of 40. 0% of 60. 50% of 4 4. 5% of 00 5. 75% of 6 6. % of 00 7. 50% of 8 8. 5% of 00 9. % of,000 Solve the problem. 0. Last summer, Jan gave away 5% of the tomatoes she raised in her garden. Jan picked a total of 0 tomatoes. How many did she give away? AGS Publishing. Permission is granted to reproduce for classroom use only
Using Percents Chapter 4, Lesson 40 This chart displays population data of the world s ten largest cities. Population of the World s Ten Largest Cities Rank City/Country Population 994 Projected Population 05 Tokyo, Japan 6,58,000 8,700,000 New York City, U.S. 6,7,000 7,600,000 São Paulo, Brazil 6,0,000 0,800,000 4 Mexico City, Mexico 5,55,000 8,800,000 5 Shanghai, China 4,709,000,400,000 6 Bombay (Mumbai), India 4,496,000 7,400,000 7 Los Angeles, U.S.,,000 4,00,000 8 Beijing, China,00,000 9,400,000 9 Calcutta, India,485,000 7,600,000 0 Seoul, South Korea,45,000,00,000 The chart shows the 994 population in each city and a projection of the population in the year 05. Use the chart to answer these questions.. In which city is the population expected to increase the most? By how many people is the population expected to increase?. In which city is the population expected to increase the least? By how many people is the population expected to increase?. Which city is expected to experience the greatest percent of increase? By what percent, to the nearest whole number, is the population expected to increase? 4. Which city is expected to experience the least percent of increase? By what percent, to the nearest whole number, is the population expected to increase? 5. In 996, the population of the world was 5,77,98,000 people. By 00, the world s population is projected to increase to 7,60,786,000 people. By what percent, to the nearest whole number, is the population of the world expected to increase between 996 and 00? AGS Publishing. Permission is granted to reproduce for classroom use only
Solving Distance, Rate, and Time Problems Chapter 4, Lesson 4 4 The distance formula is d = rt. d stands for distance, r for rate of speed, and t for time. Find d when r = 0 kilometers per hour (km/h) and t = hours. Solve: d = (0)() = 40 kilometers Use r = d t to solve for rate of speed. Find r when d = miles and t = hours. Solve: r = = miles per hour (mph) Use t = d r to solve for total time. Find t when d = 450 kilometers and r = 90 km/h. Solve: t = 4 50 = 5 hours 90 Use the appropriate version of the distance formula to find the unknown value.. d =? r = 5 mph t = hour Answer in miles.. d =? r = 8 km/h t = hours Answer in kilometers.. d = 90 miles r = 60 mph t =? Answer in hours. 4. d =,968 kilometers r =? t = 4 hours Answer in km/h. 5. d = 54 miles r = 8 mph t =? Answer in hours. 6. d =? r = 7 km/h t = hour Answer in kilometers. 7. d = kilometers r =? t = 4 hours Answer in km/h. 8. d = 4 miles r = 70 mph t =? Answer in hours. 9. d =, miles r =? t = hours Answer in mph. 0. d =? r = 96 km/h t = hour Answer in kilometers. 4 AGS Publishing. Permission is granted to reproduce for classroom use only
Using a Common Unit Cents Chapter 4, Lesson 5 4 In algebra equations, represent money as cent values. penny = cent nickel = 5 cents dime = 0 cents quarter = 5 cents dollar = 00 cents Change dollar amounts into cents by multiplying by 00. Suppose you have $.00. How many cents do you have?.00 00 = 00 cents To represent money amounts, multiply the number of coins by value. Suppose you have n nickels + twice as many (n) dimes. What is the money value? n(5) + n(0) or 5n + 0n Write the value of each amount of money in cents.. $.9. $7.46. $.00 4. $.5 5. $.95 6. $0.89 For each group of coins, choose the expression that gives the total money value of the coins in cents. Circle the letter of your choice. 7. n quarters plus twice as many dimes a. n(5) + n(0) b. n + n 8. n nickels plus half as many quarters a. n + n b. n(5) + n(5) 9. n dimes plus 4 times as many pennies a. n() + 4n(0) b. n(0) + 4n() Write an expression for the money value of the given coins, in cents. 0. n dimes + as many quarters 5 AGS Publishing. Permission is granted to reproduce for classroom use only
Calculating Simple Interest Chapter 4, Lesson 6 4 Use the formula I = prt to calculate simple interest. How much interest will $00 at 6% interest earn in year? I = $00(0.06)() = $8 Use the formula p = r to solve for p and calculate principal. It Find the principal in an account that has a rate of 5% and earns $75 interest in years. $ 75 p = (0. = $750 05)() I Use the formula r = p to solve for r and calculate rate of interest. t What is the rate of interest if $,000 earns $40 interest in years? $ 40 r = ($ = 0.08, or 8%,000)() Calculate interest on the given principal, rate, and time.. p = $00 r = % t = year. p = $70 r = 4% t = year. p = $,400 r = 7% t = years Calculate principal from the given interest, rate, and time. 4. I = $90 r = 6% t = years 5. I = $7 r = % t = year 6. I = $56 r = 8% t = year Calculate rate from the given interest, principal, and time. 7. I = $6 p = $700 t = year 8. I = $4 p = $50 t = 4 years 9. I = $48 p = $960 t = year Solve the problem. 0. For years, Will kept $655 in a savings account that earned 4% annual interest. How much interest did Will earn? AGS Publishing. Permission is granted to reproduce for classroom use only
Deriving a Formula for Mixture Problems Chapter 4, Lesson 7 44 Use this formula to determine the cost of a mixture: cost of total mixture Price per pound = number of pounds Peanuts cost $.00 per pound. Cashews cost $6.00 per pound. Suppose you mix 4 pounds of peanuts with pounds of cashews. What will the mixture cost, per pound? Price per pound = 4($ )+ ($6) 4 + = $ 4 6 = $4 Use the information in each table to answer the questions that follow it. The formula you will need is in the example on this page. Item Cost per pound Number of pounds beans $.00 7 dried tomatoes $6.00. Fill in the formula with this data.. Find the cost of this mixture. Item Cost per pound Number of pounds popcorn $9.00 peanuts $.00. Fill in the formula with this data. 4. Find the cost of this mixture. Solve the problem. 5. Suppose a grocery store mixes 4 pounds of oat cereal with pound of almonds. The oat cereal costs $.0 per pound, and the almonds cost $4.80 per pound. What should the mixture cost per pound? AGS Publishing. Permission is granted to reproduce for classroom use only
Ratio and Proportion Chapter 4, Lesson 8 45 = 4 because the cross products are equal. 6 4 6 = 4, so = 6 Find the missing term in the proportion 5 = x by making an 0 equation from the cross products. Then solve the equation. 5 x 0 0 = 5x or 5x = 0 x = 4 Therefore, 5 = 4 0 Tell whether each equation is a proportion. Prove your answer.. 4 = 8. 7 = 4. 7 = 4 4. 4 5 = 5 0 5. 4 = 6 6 6. = 0 0 7. = 6 8. 6 = 8 9. = x 5 Find the missing term in each proportion. x = 0. 4 = 6 x x =. 9 = x 7 x =. x = 0 x =. = 4 x x = 4. 7 = x 4 x = x 5. 5 = 5 x = AGS Publishing. Permission is granted to reproduce for classroom use only
Exponents Chapter 5, Lesson 46 To multiply terms with exponents, add the exponents. n n = n + = n 4 To raise a power to a power, multiply the exponents. (n ) = n = n 6 To divide terms with exponents, subtract the exponents. n 5 n = n 5 = n (Note: n 0.) Use the rule for dividing terms with exponents to find each answer.. 4 4. y 5 y, y 0. m 8 m 4, m 0 4. r 9 r, r 0 5. 6 5 6 6. (j + k) 4 (j + k),(j + k) 0 7. 0 8 0 8 8. w 7 w 7, w 0 Answer the questions to solve the problem. A square has a side s that is m long. The formula for area of a square is A = s. Fill in the blank to show how to calculate the area of this square. s = m s 9. Area = ( ) square m Next calculate the area of the square by using the rule for raising a power to a power. (See the previous answer.) 0. Area = square m
Negative Exponents Chapter 5, Lesson 47 Rewrite 0 with a negative exponent: 0. Rewrite x with a negative exponent, x 0: x. Rewrite 5 with a positive exponent: 5. Rewrite n with a positive exponent, n 0: n. 8 Rewrite using a negative exponent... 6. c, c 0 7. 8 8 5. 8. y 7, y 0 0 5 4. 0 0 6 5. 9. 8 (j k) 4 0. Rewrite using a positive exponent.. 5. 0. x 5 4. 8 5 5. y 7 6. 0 7 7. (d + k) 8. n 4 9. 7 0. (m n) Simplify each power of.. =. = 4. = 5. =. 0 =
Exponents and Scientific Notation Chapter 5, Lesson 48 Write 500.7 and 0.004 in scientific notation. Step Step Step 500.7 Count decimal places: to the left. 5.007 0 (Rule: Use a positive exponent if the decimal point moved left.) 0.004 Count decimal places: to the right. 4. 0 (Rule: Use a negative exponent if the decimal point moved right.) Write each number in scientific notation.. 9,900. 0.006. 0.099 4. 88 5.,000 6.,0,000 7. 0.00 8. 0.00004 9. 0.0009999 0. 04,9. 7,50. 9,89,50. 0.00000000 4.,000,000,000 5..6 6. 0.000000 7. 60,000,000 8. 0.00007 Solve the problems. 9. A certain bacteria cell is 0.0008 mm thick. Write this measurement in scientific notation. 0. Dinosaurs became extinct (that is, died out) about 65 million years ago. This number is written out as 65,000,000. Rewrite it in scientific notation.
Computing in Scientific Notation Chapter 5, Lesson 4 49 Multiply: 0.005 7,50,000 Step 0.005 = 5.0 0 7,50,000 = 7.5 0 6 Step (5.0 0 )(7.5 0 6 ) = (5.0 7.5)(0 0 6 ) Step (5.75)(0 + 6 ) = (5.75)(0 ) Step 4 (.575)(0)(0 ) = (.575)(0 + ) = (.575)(0 4 ) Find each product. Write the answer in scientific notation.. 0.0007 90.,400,000 0.006. 0.008 0.054 4. 8,500,50 5.,600 0.00000005 Divide: 0,000,000 50 Step 0,000,000 =. 0 8 50 =.5 0. 0 8.5 0 Step = (..5)(0 8 0 ) Step (0.44)(0 8 ) = 0.44(0 6 ) Step 4 4.4(0 )(0 6 ) = (4.4)(0 + 6 ) = (4.4)(0 5 ) Find each quotient. Write the answer in scientific notation. 6. 9,000,000,00 7. 7,700 0.005 8.,008 900,000 9. 0.00086 0.05 0. 9.8 0.000
Defining and Naming Polynomials Chapter 5, Lesson 5 50 The chart summarizes the kinds of polynomials. The greatest power of a variable is called the degree of a polynomial. Expression Name of the Polynomial Degree y monomial y + 5 binomial y y + 5 trinomial y + y y + 5 polynomial Fill in the missing data in the chart. Write on each numbered blank. Expression Name of the Polynomial Degree n + n. k k + k 4 polynomial. 5x monomial. x 4. 7y + 4y 5 trinomial 5. n + n 8n 8 6. 7. 9n 8. k k + 7 9. 0. b + 4.. Each expression is described incorrectly. Write what is wrong with the description.. k + 5 binomial in k, degree 4. y 4 trinomial in y, degree 5. r r r + 7 polynomial in x, degree
Adding and Subtracting Polynomials Chapter 5, Lesson 6 5 Add (x + 4x + 8) and (x x x + ). x + 4x + 8 + x x x + x + x x + Subtract (x 4x ) from (4x + 7x x + 4). Find the opposite of the expression to be subtracted: ( )(x 4x ) = x + 4x + Add: 4x + 7x x + 4 + x + 4x + x + 7x + x + 5 Find each sum.. (5k 9k + k ) and (k + 0k + k + 4). (y 4 + y 7y) and (5y y + y + 9) Find each difference. Remember to add the opposite.. (4n 8n + ) (n + 5n 4) 4. (6x 4 x + x 6) (5x + ) Solve the problem. 5. A nut company will close one of its two stores and combine all of the inventory (the nuts in stock) from the two stores. The following polynomials give the number of bags of dry-roasted peanuts in each store: Lakewood store: x + 4x Downtown store: x x + 4 Find the combined inventory of the dry-roasted peanuts.
Multiplying Polynomials Chapter 5, Lesson 7 5 Multiply (n + )(n ). Use the distributive property. (n + )(n ) = n(n ) + (n ) = n n + n 6 = n n 6 Find each product.. (k + 4)(k 6). ( x + 5)( x 0). (c + c) 4. (d )(d + 5) 5. (y + 9)(y y + 6) 6. (n )(n + 6n ) 7. (w + )(w 4) 8. ( x 5)(x 5 + x 4 4x + 6x ) 9. (c )(c c + 9) 0. (k + 7)(k 5). (m )(m 4 m 7m ). (8b + )(b 9). (k )(k 6 + k 5 k 4 8k + 4k 9k 9) 4. (7n 4)(n 8n 5n + 8) Solve the problem. 5. A machine in a factory turns out a large metal grid (a crisscross or checkerboard pattern), which later gets cut into small pieces for computer parts. The measurements of this large grid are as follows: length: (x + ) width: (x ) Find the area of this grid, using the formula: Area = length width.
Special Polynomial Products Chapter 5, Lesson 8 5 Each of these polynomial products forms a pattern. (a + b) = (a + b)(a + b) = a(a + b) + b(a + b) = a + ab + b (a b) = (a b)(a b) = a(a b) b(a b) = a ab + b (a + b)(a b) = a(a b) + b(a b) = a b (a + b) = (a + b)(a + b)(a + b) = (a + b)[(a + b)(a + b)] = a + a b + ab + b Study each product. Decide what polynomials were multiplied to give the product. Use the example above as a guide. Write your answer in the blank.. m + mn +n. j k. x + xy +y 4. c + c d + cd + d 5. w x 6. g gh + h Find each product. Compare your solutions with the patterns in the example above. 7. (p + r)(p r) 8. (f + g) 9. (y + z) 0. (a + d)(a d). (t + u)(t + u)(t + u). (n + p)(n + p) 4. (c d) 5. (k + m)(k m). (p q)(p q)
Exponents and Complex Fractions Chapter 5, Lesson 8 54 It is possible to simplify complex fractions that contain exponents. Simplify. a b a Step Rewrite the complex fraction horizontally. Recall that the fraction bar separating the numerator from the denominator means "divide." a b a a = b a Step Multiply each term in the expression by the reciprocal of the divisor. a b a b a = ( a )( b a ) ( )( ) b a Step Simplify. ( a )( b a ) (b a a )( a b ) = ( a )( b ) = ( a b ) Simplify these complex fractions.. b b a 4. g g 7. 4c de 4 d 8 e d 0. 8c 9 5 b d cd b. c d 5. x zy x y z 8. r st u r u. a 5 5 b 6. 0m n p p n 9. g 4hi ah g h a i
Dividing a Polynomial by a Monomial Chapter 5, Lesson 9 55 Find the quotient of (x 6x + 9x) x. Step Rewrite the problem. Step Divide each term of the numerator by the term in the denominator. x x 6 x x 9x x x 6x + 9x x = x x + (quotient) Step Check by multiplying quotient by divisor. The answer should be the dividend. (x x + )(x) = x 6x +9x (dividend) Find each quotient. Check your work using multiplication.. (5n 5n + 45) 5. (6y 4y) (4y). (k 7 4k 6 + 6k 5 k 6k ) (k ) Solve the problems. 4. An engineer in a paper-clip factory represents the number of paper clips that come out of a machine in one hour by the following polynomial expression: x 5 + 4x 4 + 6x 8x. The paper clips are packed in boxes, each of which holds x paper clips. How many boxes will be filled by an hour s run of the paper-clip machine? 5. A textile factory produces a bolt (roll) of cloth 40 yards long. The expression 6k 4 + 8k + 4k k gives the number of threads in this bolt of cloth. If the bolt is cut into 8 equal pieces of cloth, how many threads will each piece have?
Dividing a Polynomial by a Binomial Chapter 5, Lesson 0 56 Find the quotient of (x x 5) (x + ). x 4 x + x x 5 (x + x) 4x 5 ( 4x 4) remainder Check by multiplication: (x + )(x 4) = x x 5 Find each quotient. Identify any remainder.. (x x 5) (x + 5). (4a 6a 4) (7a + ). (5y + 4y ) (y + ) 4. (d d 5) (d + ) 5. x 4 x + x 5 x 5 + x 4? Tell what is wrong with the following division work. Show how to correct the error. + x 9
Polynomials in Two or More Variables Chapter 5, Lesson 57 Evaluate P(x, y) = x + xy + y for x = and y = Step Substitute the variables with their values. P(, ) = () + ()( ) + ( ) Step Step 4 + 4 + 4 Follow the order of operations. Add. 4 + 4 + 4 = 4 P(x, y) = 4 Evaluate P(x, y) = x + xy + y for each set of values.. x =, y =. x =, y = 6. x =, y = 4. x = 6, y = 5 5. x =, y = 8 Evaluate P(x, y) = x y + x y + xy at 6. P(, ) 7. P(, ) 8. P(7, 0) 9. P(8, ) 0. P(, 5) Evaluate P(x, y, z) = x yz + x y z + xy + yz for. x =, y =, z =. x =, y = 0, z = 4. x =, y =, z = 4. x = 0, y = 5, z = 5. x =, y = 4, z = 0
Greatest Common Factor Chapter 6, Lesson 58 Find the GCF. 40 and 56 49k 4 and k Step Write the factorizations. 40 = 5 7 56 = 7 49k 4 = 7 7 k k k k k = 7 k k Step Identify common prime factors. 7 7 k k Step Write the GCF as a product. 7 = 8 7k Find the GCF for these groups of integers.. 60, 6. 6, 70. 45, 5 4. 64, 4 5. 4, 90 Find the GCF for these groups of expressions. 6. 4x 5 y 4,7xy 7. j k 4,54j k 6 8. 4a b,8a b 9. 5m 6 n,0m 5 n Solve the problem. 0. Dad just had a birthday. Before this birthday, dividing Dad s age by left a remainder of. How do you know that Dad s new age is not a prime number?
Factoring Polynomials Chapter 6, Lesson 59 Factor 5x y 4x y. Step Find the GCF: 5x y = 5 7 x x x y y 4x y = 7 x x y y y The GCF is 7x y. Step Rewrite the expression using the GCF. 5x y 4x y = 7x y (5x)() 7x y ()()(y) = 7x y (5x y) by the distributive property Step Check. 7x y (5x y) = 5x y 4x y Find the GCF and factor these expressions.. 6a + 9a. b 4 4b. 4d + 8d cd 4. 6x 9x y 5. a ab + 9a b 6. j k jk 7. xyz 8xy z 8. m n p + 6m np mp Solve the problems. 9. Suppose you have the following in your fruit bowl: x apples 6x peaches x pears In all, you have x + 7x pieces of fruit. Factor this expression. 0. With the same contents in your fruit bowl, suppose you eat all of the x apples? Write an expression to represent the fruit you will now have left. Can this expression be factored? If it can, factor it.
Factoring Trinomials: x + bx + c Chapter 6, Lesson 60 Factor a + a 5. Step a + a 5 = ( + )( ) Step a + a 5 = (a + )(a ) to give a Step Find factors of 5 whose sum is. (5)( ) = 5, and (5) + ( ) = a + a 5 = (a + 5)(a ) Step 4 Check by multiplying. (a + 5)(a )= a + a 5 Factor the expressions. Check by multiplying.. y + 7y +. w 4w. b 9b + 4 4. x x + 8 5. n 7n + 6. z z 0 7. d + 5d + 6 8. a 5a 50 9. m m 5 Solve the problem. 0. A grid (checkerboard pattern) is printed on each sheet of graph paper produced in a paper factory. The total number of squares on the printed grid is x 6x 7. What is the length, in squares, of each side of the grid? (Hint: factor the trinomial.)
Factoring Trinomials: ax + bx + c Chapter 6, Lesson 4 6 Factor x + x + 4. Step x + x + 4 = ( x + )( x + ) to give x Step Find factors of and 4 whose sum is. Factors of = ()() Factors of 4 = ()()() After trying out the possible combinations, the following factors of the trinomial are found: (x + )(x + 4) Step Check by multiplying. (x + )(x + 4) = x + x + 4 Factor these expressions.. a + 4a +. x 4x 4. 6d d 5 4. 8x 8x + 9 5. 4n + n + 6. y 7y + 7. 4x + x 8. n 5n + 9. 6b + 7b 0 Solve the problem. 0. A pretzel-maker fills identical bags with an equal number of pretzels. In one hour, the pretzel-maker bags (4k + 7k + 8) pretzels in all. Factor this trinomial to find the number of bags (larger factor) and the number of pretzels per bag (smaller factor).