School District of Palm Beach County. Summer Packet Algebra EOC Review Answers

Transcription

1 School District of Palm Beach Count Summer Packet Algebra EOC Review Answers

2 - Reteaching Variables and Epressions You can represent mathematical phrases and real-world relationships using smbols and operations. This is called an algebraic epression. For eample, the phrase 3 plus a number n can be epressed using smbols and operations as 3 n. What is the phrase minus a number d as an algebraic epression? minus a number d d The phrase minus a number d, rewritten as an algebraic epression, is d. The left side of the table below gives some common phrases used to epress mathematical relationships, and the right side of the table gives the related smbol. Phrase sum difference product quotient less than more than Smbol 3 Write an algebraic epression for each word phrase.. plus a number d d. the product of and g 3 g 3. fewer than a number f f. 7 less than h h 7. the quotient of 0 and t 0 t 6. the sum of and Write a word phrase for each algebraic epression. 7. h 6 8. m 9. q 3 0 the sum of h and 6 less than a number m the product of q and r. h m. n the quotient of 3 and r the sum of h and m the product of and n 9

3 - Reteaching (continued) Variables and Epressions Multiple operations can be combined into a single phrase. What is the phrase minus the product of 3 and d as an algebraic epression? minus the product of 3 and a number d 3 3 d The phrase minus the product of 3 and a number d, rewritten as an algebraic epression, is 3d. Write an algebraic epression for each phrase. 3. less than the quotient of and a number z z. greater than the product of 3 and a number q 3 3 q. the quotient of h and n 3 h n 3 6. the difference of 7 and t 7 t Write an algebraic epression or equation to model the relationship epressed in each situation below. 7. Jane is building a model boat. Ever inch on her model is equivalent to 3. feet on the real boat her model is based on. What would be the mathematical rule to epress the relationship between the length of the model, m, and the length of the boat, b? 3.m b 8. Ln is putting awa savings for his college education. Ever time Ln puts mone in his fund, his parents put in $. What is the epression for the amount going into Ln s fund if Ln puts in L dollars? L 0

4 - Reteaching Order of Operations and Evaluating Epressions Eponents are used to represent repeated multiplication of the same number. For eample, The number being multiplied b itself is called the base; in this case, the base is. The number that shows how man times the base appears in the product is called the eponent; in this case, the eponent is. is read four to the fifth power. How is written using an eponent? The number 6 is multiplied b itself 7 times. This means that the base is 6 and the eponent is written using an eponent is 6 7. Write each repeated multiplication using an eponent (7) 3 (7) 3 (7) 3 (7) (7) 3 Write each epression as repeated multiplication Q 7 R Q 7 R 3 Q 7 R 3 Q 7 R 3 Q 7 R. 7??????. (n) n 3 n 3 n 3 n 3 n 3. Trisha wants to determine the volume of a cube with sides of length s. Write an epression that represents the volume of the cube. s 3 9

5 - Reteaching(continued) Order of Operations and Evaluating Epressions The order of operations is a set of guidelines that make it possible to be sure that two people will get the same result when evaluating an epression. Without this standard order of operations, two people might evaluate an epression differentl and arrive at different values. For eample, without the order of operations, someone might evaluate all epressions from left to right, while another person performs all additions and subtractions before all multiplications and divisions. You can use the acronm P.E.M.A. (Parentheses, Eponents, Multiplication and Division, and Addition and Subtraction) to help ou remember the order of operations. How do ou evaluate the epression 3 3 0? There are no parentheses or eponents, so first, 3 8 do an multiplication or division from left to right. Do an addition or subtraction from left to right. 9 Simplif each epression.. ( 3) 6. (8 )( 6) 6. ( 3) 3 7. Q 3 R ( ) Write and simplif an epression to model the relationship epressed in the situation below.. Manuela has two boes. The larger of the two boes has dimensions of cm b cm b 0 cm. The smaller of the two boes is a cube with sides that are 0 cm long. If she were to put the smaller bo inside the larger, what would be the remaining volume of the larger bo? cm 3 0

6 -3 Reteaching Real Numbers and the Number Line A number that is the product of some other number with itself, or a number to the second power, such as , is called a perfect square. The number that is raised to the second power is called the square root of the product. In this case, 3 is the square root of 9. This is written in smbols as!9 3. Sometimes square roots are whole numbers, but in other cases, the can be estimated. What is an estimate for the square root of 0? There is no whole number that can be multiplied b itself to give the product of You cannot find the eact value of!0, but ou can estimate it b comparing 0 to perfect squares that are close to 0. 0 is between and 69, so!0 is between! and!69.!,!0,!69,!0, 3 The square root of 0 is between and 3. Because 0 is closer to than it is to 69, we can estimate that the square root of 0 is slightl greater than. Find the square root of each number. If the number is not a perfect square, estimate the square root to the nearest integer A square mat has an area of cm. What is the length of each side of the mat? cm 9

7 -3 Reteaching (continued) Real Numbers and the Number Line The real numbers can be separated into smaller, more specific groups, called subsets. Each of these subsets has certain characteristics. For eample, a rational number can be epressed as a fraction of two integers, with the denominator of the fraction not equal to 0. Irrational numbers cannot be epressed as a fraction of two integers. Ever real number belongs to at least one subset of the real numbers. Some real numbers belong to multiple subsets. To which subsets of the real numbers does 7 belong? 7 is a natural number, a whole number, and an integer. But 7 is also a rational number because it can be written as 7, a fraction of two integers with the denominator not equal to 0. A number cannot belong to both the subset of rational numbers and the subset of irrational numbers, so 7 is not an irrational number. List the subsets of the real numbers to which each of the given numbers belongs !3 rational, whole, rational, whole, irrational natural, integer natural, integer rational rational, integer rational, integer rational !6 rational, whole, integer rational, natural, whole, integer 0.!0.! irrational rational rational 30

8 - Reteaching Properties of Real Numbers Equivalent algebraic epressions are epressions that have the same value for all values for the variable(s). For eample and are equivalent epressions since, regardless of what number is substituted in for, simplifing each epression will result in the same value. Certain properties of real numbers lead to the creation of equivalent epressions. Commutative Properties The commutative properties of addition and multiplication state that changing the order of the addends does not change the sum and that changing the order of factors does not change the product. Addition: a b b a Multiplication: a? b b? a To help ou remember the commutative properties, ou can think about the root word commute. To commute means to move. If ou think about commuting or moving when ou think about the commutative properties, ou will remember that the addends or factors move or change order. Do the following equations illustrate commutative properties? a. 3 3 b. ( 3 3) 3 3 (3 3 ) c and 3 both simplif to 7, so the two sides of the equation in part (a) are equal. Since both sides have the same two addends but in a different order, this equation illustrates the Commutative Propert of Addition. The epression on each side of the equation in part (b) simplifies to 30. Both sides contain the same 3 factors. However, this equation does not illustrate the Commutative Propert of Multiplication because the terms are in the same order on each side of the equation. 3 and 3 do not have the same value, so the equation in part (c) is not true. There is not a commutative propert for subtraction. Nor is there a commutative propert for division. Associative Properties The associative properties of addition and multiplication state that changing the grouping of addends does not change the sum and that changing the grouping of factors does not change the product. Addition: (a b) c a (b c) Multiplication: (a? b)? c a? (b? c) 39

9 - Reteaching (continued) Properties of Real Numbers Do the following equations illustrate associative properties? a. ( ) ( ) b. 3 ( 3 7) 3 (7 3 ) ( ) and ( ) both simplif to 0, so the two sides of the equation in part (a) are equal. Since both sides have the same addends in the same order but grouped differentl, this equation illustrates the Associative Propert of Addition. The epression on each side of the equation in part (b) simplifies to 6. Both sides contain the same 3 factors. However, the same factors that were grouped together on the left side have been grouped together on the right side; onl the order has changed. This equation does not illustrate the Associative Propert of Multiplication. Other properties of real numbers include: a. Identit propert of addition: a b. Identit propert of multiplication: a? a 3? 3 c. Zero propert of multiplication: a? 0 0 6? 0 0 d. Multiplicative propert of negative one:? a a? 7 7 What propert is illustrated b each statement?. (m 7.3). m (7.3.). p? p Associative Propert of Addition Multiplicative Identit (3r)(s) (s)(3r) Additive Identit Commutative Propert of Multiplication. 7 () () 7 6. (3) 3 Commutative Propert of Addition Multiplicative Propert of Simplif each epression. Justif each step. 7. ( 8) 3 8. (? m)? 7 (8 ) 3 Comm. Prop. of Add. 8 ( 3) Assoc. Prop. of Add. 8 Combine like terms. 9. (7 7) 0 Add. Ident. Simplif. (m? )? 7 Comm. Prop. of Mult. m? (? 7) Assoc. Prop. of Mult. 3m Comm. Prop. of Mult. 0

10 - Reteaching Adding and Subtracting Real Numbers You can add real numbers using a number line or using the following rules. Rule : To add two numbers with the same sign, add their absolute values. The sum has the same sign as the addends. What is the sum of 7 and? Use a number line The sum is. Start at zero. Move 7 spaces to the left to represent 7. Move another spaces to the left to represent. Use the rule. 7 () The addends are both negative. 7 Add the absolute values of the addends and. 7 () The sum has the same sign as the addends. Rule : To add two numbers with different signs, subtract their absolute values. The sum has the same sign as the addend with the greater absolute value. What is the sum of 6 and 9? Use the rule. 9 (6) The addends have different signs. 9 6 Subtract the absolute values of the addends and (6) 3 The positive addend has the greater absolute value. 9

11 - Reteaching (continued) Adding and Subtracting Real Numbers Find each sum (7) 6. 8 () 6 6. () (.) Addition and subtraction are inverse operations. To subtract a real number, add its opposite. What is the difference (8)? (8) 8 The opposite of 8 is 8. 3 Use Rule. The difference (8) is 3. Find each difference () () 6. () (.) (.) The temperature was 8C. Five hours later, the temperature had dropped 08C. What is the new temperature? 8C 0. Reasoning Which is greater, (77) or (77)? Eplain. (77) is greater. It is the same as 77 which is a positive number. The sum of (77) is a negative number. 0

12 -6 Reteaching Multipling and Dividing Real Numbers You need to remember two simple rules when multipling or dividing real numbers.. The product or quotient of two numbers with the same sign is positive.. The product or quotient of two numbers with different signs is negative. What is the product 6( 30)? 6(30) 80 6 and 30 have the same sign so the product is positive. What is the quotient 7 (6)? 7 (6) 7 and 6 have different signs so the quotient is negative. Find each product or quotient.. (6) 30. 7(0) (9) 9 6. () ? () (6) () 0.7..() Q 3 R. 3? Q 3 R 6. The temperature dropped F each hour for 6 hours. What was the total change in temperature? 8 F 7. Reasoning Since and (), what are the two values for the square root of? and 9

13 -6 Reteaching (continued) Multipling and Dividing Real Numbers The product of 7 and 7 is. Two numbers whose product is are called reciprocals. To divide a number b a fraction, multipl b its reciprocal. What is the quotient 3 Q 7 R? 3 Q 7 R 3 3 Q 7 R To divide b a fraction, multipl b its reciprocal. The signs are different so the answer is negative. Find each quotient Q 3 R 3. Q R. Q 7 R Q R Writing Another wa of writing a b is a b. Eplain how ou could evaluate. 6 What is the value of this epression? Change the problem to the equivalent division problem 6. To find this quotient, change this division problem to the multiplication problem 3 6. The answer is 3. 60

14 -7 Reteaching The Distributive Propert The Distributive Propert states that the product of a sum and another factor can be rewritten as the sum of two products, each term in the sum multiplied b the other factor. For eample, the Distributive Propert can be used to rewrite the product 3( ) as the sum 3 3. Each term in the sum is multiplied b 3; then the new products are added. What is the simplified form of each epression? a. ( ) b. ( 3)(3) () () Distributive Propert (3) 3(3) Distributive Propert 0 Simplif. 6 9 Simplif. The Distributive Propert can be used whether the factor being multiplied b a sum or difference is on the left or right. The Distributive Propert is sometimes referred to as the Distributive Propert of Multiplication over Addition. It ma be helpful to think of this longer name for the propert, as it ma remind ou of the wa in which the operations of multiplication and addition are related b the propert. Use the Distributive Propert to simplif each epression.. 6(z ). ( k) 3. ( ). (7 n)0 6z k n. (3 8w). 6. (p ).6 7. ( ) 8. 6(q ) 3. 36w Write each fraction as a sum or difference. 9. m z m z Simplif each epression.. f 9 8f 3 3. d 6 6 d (6 j). (9h ). (n ) 6. (6 8f) 6 j 0.p 3 9h 6 n 6q 6 8f 69

15 -7 Reteaching (continued) The Distributive Propert The previous problem showed how to write a product as a sum using the Distributive Propert. The propert can also be used to go in the other order, to convert a sum into a product. How can the sum of like terms 6 be simplified using the Distributive Propert? Each term of 6 has a factor of. Rewrite 6 as () 6(). Now use the Distributive Propert in reverse to write () 6() as ( 6), which simplifies to. Simplif each epression b combining like terms n 7n 9. p 6p 8 8n 0. a 9a. 9k k. t 0t a k 8t B thinking of or rewriting numbers as sums or differences of other numbers that are easier to use in multiplication, the Distributive Propert can be used to make calculations easier. p How can ou multipl 78 b 0 using the Distributive Propert and mental math? Write the product (00 ) Rewrite 0 as sum of two numbers that are eas to use in multiplication. 78(00) 78() Use the Distributive Propert to write the product as a sum Multipl Simplif. Use mental math to fintd each product ,

16 -8 Reteaching An Introduction to Equations An equation is a mathematical sentence with an equal sign. An equation can be true, false, or open. An equation is true if the epressions on both sides of the equal sign are equal, for eample 3. An equation is false if the epressions on both sides of the equal sign are not equal, for eample. An equation is considered open if it contains one or more variables, for eample 8. When a value is substituted for the variable, ou can then decide whether the equation is true or false for that particular value. If an open sentence is true for a value of the variable, that value is called a solution of the equation. For 8, 6 is a solution because when 6 is substituted in the equation for, the equation is true: 6 8. Is the equation true, false, or open? Eplain. a The equation is true, because both epressions equal 36. b. 8? The equation is false, because 8 3 and? ; 3. c. n The equation is open, because there is a variable in the epression on the left side. Tell whether each equation is true, false, or open. Eplain.. () 3(6) false; () 3(6) 6 open; it contains a variable true. (8) (8 ) 0 false; (8) open; it contains a variable true 7. (8) 8 8. (3 6) true false; (3 6) 3 true Is 3 a solution of the equation 7? 7 (3) 7 Substitute 3 for. 7 7 Simplif. Since 7 7, 3 is a solution of the equation 7. Tell whether the given number is a solution of each equation. 0. 7; 7. 8 n ;. 3p ; 9 no es no 3. k (6)(8) ; 6. 0v 36 6; ; no no es 6. 7t 8; m ; 7 8. g 8 3 ; 38 es no es 79

17 -8 Reteaching (continued) An Introduction to Equations A table can be used to find or estimate a solution of an open equation. You will have to choose a value to begin our table. If ou choose the value that makes the equation true, ou have found the solution and are done. If our choice is not the solution, make another choice based on the values of both sides of the equation for our first choice. If ou choose one value that makes one side of the equation too high and then another value that makes that same side too low, ou know that the solution must lie between the two values ou chose. It ma not be possible to determine an eact solution for each equation; estimating the solution to be between two integers ma be all that is possible in some cases. What is the solution of 6n 8 8? If n, then the left side of the equation is 6() 8 or 0, which is too low. If n, then the left side of the equation is 6() 8 or 38, which is too high. The solution must lie between and, so keep tring values between them. If n 3, then the left side of the equation is 6(3) 8 or 6, which is too low. If n, then the left side of the equation is 6() 8 or 3, which is too high. The solution must lie between 3 and, but there are no other integers between 3 and. You can give an estimate for the solution of 6n 8 8 as being between the integers 3 and. Write an equation for each sentence times the sum of a number and is 9. 3(n ) 9 0. Negative 8 times a number minus is equal to 30. 8n 30. Jared receives $3 for each lawn he mows. What is an equation that relates the number of lawns w that Jared mows and his pa p? p 3w. Shariff has been working for a compan ears longer than Pats. What is an equation that relates the ears of emploment of Shariff S and the ears of emploment of Pats P? S P Use mental math to find the solution of each equation. 3. h n 3. 6 k t 3 7. z 8. j c a 3 Use a table to find the solution of each equation. 3. 3b t

18 -9 Reteaching Patterns, Equations, and Graphs Tables, equations, and graphs are some of the was that a relationship between two quantities can be represented. You can use the information provided b one representation to produce one of the other representations; for eample, ou can use data from a table to produce a graph. You can also use an of the representations to draw conclusions about the relationship. Are (, ) and (, 3) solutions of the equation 3? For each ordered pair, ou can substitute the - and - coordinates into the equation for and and then simplif to see if the values satisf the equation. For (, ): For (, 3): 3() Substitute for and. 3 3() Multipl and then add. 3 0 Since both sides of the equation have the same value, the ordered pair (, ) is a solution of the equation 3. Since the two sides of the equation have different values, the ordered pair (, 3) is not a solution of the equation 3. The table shows the relationship between the number of hours Kaa works at her job and the amount of pa she receives. Etend the pattern. How much mone would Kaa earn if she worked 0 hours? Method : Write an equation..0 Kaa earns $.0 per hour..0(0) Substitute 0 for. 00 Simplif. She would earn $00 in 0 hours. Hours Worked Mone Earned ($) Method : Draw a graph. She would earn $00 in 0 hours. Mone Earned ($) O Hours worked 89

19 -9 Reteaching (continued) Patterns, Equations, and Graphs Tell whether the equation has the given ordered pair as a solution.. 7; (, ) es. 6; (, ) no 3. ; (, 0) no. ; (3, ). 8; (7, ) 6. 3 ; (, no es ) es Use a table, an equation, and a graph to represent each relationship. 7. Tickets to the fair cost $7. 8. Brian is ears older than Sam. Tickets 3 Cost ($) Sam (rs) Brain (rs) Use the table to draw a graph and answer the question. 9. The table shows Jake s earnings for the number of cakes he baked. What are his earnings for baking 7 cakes? $800 Use the table to write an equation and answer the question. Cakes 0 Earnings ($) The table shows the number of miles that Kate runs on a weekl basis while training for a race. How man total miles will she have run after weeks? Training Weeks 3 Miles Run The table shows the amount of mone Kevin receives for items that he sells. How much will he earn if he sells 30 items? Items Earnings 0; 600 mi Sold ($) 7; $

20 - Reteaching Solving One-Step Equations You can use the properties of equalit to solve equations. Subtraction is the inverse of addition. What is the solution of 33? In the equation, 33, is added to the variable. To solve the equation, ou need to isolate the variable, or get it alone on one side of the equal sign. Undo adding b subtracting from each side of the equation. Drawing a diagram can help ou write an equation to solve the problem. Part Whole Part X 33 Solve Undo adding b subtracting. 8 Simplif. This isolates. Check 33 Check our solution in the original equation Substitute 8 for The solution to 33 is 8. Division is the inverse of multiplication. What is the solution of? In the equation,, the variable is divided b. Undo dividing b b multipling b on each side of the equation. X Solve?? Undo dividing b b multipling b. 60 Simplif. This isolates. The solution to is 60. 9

21 - Reteaching (continued) Solving One-Step Equations Solve each equation using addition or subtraction. Check our answer.. 3 n 9. f m 0. r 7. b.. 6. t 9 Define a variable and write an equation for each situation. Then solve. 7. A student is taking a test. He has 37 questions left. If the test has 78 questions, how man questions has he finished? f 37 78; 8. A friend bought a bouquet of flowers. The bouquet had nine daisies and some roses. There were a total of flowers in the bouquet. How man roses were in the bouquet? 9 r ; 6 Solve each equation using multiplication or division. Check our answer. 9. z c q a g. 0. s.. A student has been tping for minutes and has tped a total of 96 words. Write and solve an equation to determine the average number of words she can tpe per minute. a 96; 68 0

22 - Reteaching Solving Two-Step Equations Properties of equalit and inverse operations can be used to solve equations that involve more than one step to solve. To solve a two-step equation, identif the operations and undo them using inverse operations. Undo the operations in the reverse order of the order of operations. What is the solution of 8 3? To get the variable term alone on the left side, add 8 to each side. Simplif. Divide each side b since is being multiplied b on the left side. This isolates. Simplif. Check 8 3 Check our solution in the original equation. (8) 8 3 Substitute 8 for Simplif. To solve 6 3, ou can use subtraction first to undo the addition, and then use multiplication to undo the division. What is the solution of 6 3? 6 3 To get the variable term alone on the right, subtract from each side. 3 Simplif. 3() 3a 3 b Since is being divided b 3, multipl each side b 3 to undo the division. This isolates. 63 Simplif. 9

23 - Reteaching (continued) Solving Two-Step Equations Solve each equation. Check our answer.. f b 3. z 7 8. w n Solve each equation. Justif each step. 7. 6d 3 6d 3 Add to each side. (Add. Prop. of Equal.) 6d 36 Simplif. 6d Divide both sides b 6. (Div. Prop. of Equal.) d 6 Simplif. 8. p 7? p 7? Multipl both sides b. (Mult. Prop. of Equal.) p 7 0 Simplif. p Add 7 to both sides. (Add. Prop. of Equal.) p 3 Simplif. Define a variable and write an equation for each situation. Then solve. 9. Ra s birthda is 8 more than four times the number of das awa from toda than Jane s birthda. If Ra s birthda is das from toda, how man das until Jane s birthda? 8 j; in das 0. Jerud weighs pounds less than twice Kate s weight. How much does Kate weigh if Jerud weighs 0 pounds? 0 k ; 0 lb. A phone compan charges a flat fee of $7 per month, which includes free local calling plus $0.08 per minute for long distance calls. The Talor s phone bill for the month is $3.80. How man minutes of long distance calling did the use during the month? m 3.80; 8 min. A deliver compan charges a flat rate of $3 for a large envelope plus an additional $0. per ounce for ever ounce over a pound the package weighs. The postage for the package is $.0. How much does the package weigh? (Hint: remember the first pound is included in the $3.) e; lb 0 oz; 0

24 -3 Reteaching Solving Multi-Step Equations To solve multi-step equations, use properties of equalit, inverse operations, the Distributive Propert, and properties of real numbers to isolate the variable. Like terms on either side of the equation should be combined first. a) What is the solution of 3 8 3? Group the terms with together so that the like terms are grouped together. 0 8 Add the coefficients to combine like terms To get the variable term b itself on the left side, subtract 8 from each side Simplif Divide each side b 0 since is being multiplied b 0 on the left side. This isolates. Simplif. b) What is the solution of (3n ) 0? 6n 8 0 Distribute the into the parentheses b multipling each term inside b. 6n To get the variable term b itself on the left side, subtract 8 from each side. 6n 8 6n n 3 Solve each equation. Check our answer. Simplif. Divide each side b 6 since n is being multiplied b 6 on the left side. This isolates n. Simplif.. 6h 8h n 3n 3.. 8(3d ) 88. ( ) 6. 3 (k ) m 6 m 6 8. (3r ) 3r (6t ) 8 0. (0b ) ( ). 8 3(z 6) z Reasoning Solve the equation 7( ) using two different methods. Show our work. Which method do ou prefer? Eplain. 3; Check students work. 9

25 -3 Reteaching (continued) Solving Multi-Step Equations Equations with fractions can be solved b using a common denominator or b eliminating the fractions altogether. What is the solution of 3 7? Method Method Get a common denominator first. 3 3 Q R Q 3 R ? 3? 3 Multipl b the common denominator first. 3 Q 3 R Q 7 R Q R Q 3 R Q 7 R Decimals can be cleared from the equation b multipling b a power of ten with the same number of zeros as the number of digits to the right of the decimal. For instance, if the greatest number of digits after the decimal is 3, like.86, ou multipl b 000. What is the solution of.8..? 00(.8..) Multipl b 00 because the most number of digits after the decimal is two. 80 Simplif b moving the decimal point to the right places in each term Add to each side to get the term with the variable b itself on the left side. 3. Solve each equation. Check our answer. Divide each side b 80 to isolate the variable a n j w

26 - Reteaching Solving Equations With Variables on Both Sides To solve equations with variables on both sides, ou can use the properties of equalit and inverse operations to write a series of simpler equivalent equations. What is the solution of m m 3 6m? 7m 6m 9 Add the terms with variables together on the left side and the constants on the right side to combine like terms. 7m 6m 6m 9 6m 3m 9 3m 9 3m 3 3m m To move the variables to the left side, add 6m to each side. Simplif. To get the variable term alone on the left, add to each side. Simplif. Divide each side b 3 since is being multiplied b 3 on the left side. This isolates. Simplif. What is the solution of 3( ) 3( 6)? Distribute 3 on the left side and 3 on the right side into the parentheses b multipling them b each term inside Solve each equation. Check our answer. To move all of the terms without a variable to the right side, add 6 to each side. Simplif. To get the variable terms to the left side, add 3 to each side. Simplif. Divide each side b 8 since is being multiplied b 8 on the left side. This isolates. Simplif and reduce the fraction n n 8 3. (g ) g 8.. d 3d d 6. (m ) (3m 3) 6. ( 8) ( ) 0 7. a (a ) 7a 8. w (3w ) w 9. (3 p) (3p 3) 7 39

27 - Reteaching (continued) Solving Equations With Variables on Both Sides An equation that is true for ever value of the variable for which the equation is defined is an identit. For eample, is an identit because the equation is true for an value of. An equation has no solution if there is no value of the variable that makes the equation true. The equation 6 3 has no solution. What is the solution of each equation? a) 3( ) (6 3) Distribute 3 on the left side and on the right side into the parentheses b multipling them b each term inside. To get the variable terms to the left side, subtract from each side. 6 6 Simplif. Because 6 6 is alwas true, there are infinitel man solutions of the original equation. The equation is an identit. b) n (n ) 8 6n n n 8 8 6n 6n 8 8 6n 6n 8 6n 8 6n 6n 8 8 Simplif. Since 8 8, the equation has no solution. Distribute into the parentheses b multipling it b each term inside. Add the variable terms on the left side to combine like terms. To get the variable terms to the left side, subtract 6n from each side. Determine whether each equation is an identit or whether it has no solution. 0. 3( ) (3 ). (3 ) (6 8). 3n 3(n 3) 3 no solution an identit no solution Solve each equation. If the equation is an identit, write identit. If it has no solution, write no solution. 3. (n ) (n ). (d ) d 8. k 8 k 3 no solution 0 an identit 6. Open-Ended Write three equations with variables on both sides of the equal sign with one having no solution, one having eactl one solution, and one being an identit. Answers ma var. Sample: no solution: ; one solution: 3 7 ; an identit: 0 7 0; 0

28 - Reteaching Literal Equations and Formulas A literal equation is an equation that involves two or more variables. When ou work with literal equations, ou can use the methods ou have learned in this chapter to isolate an particular variable. To solve for specific values of a variable, simpl substitute the values into our equation and simplif. What is the solution of 3 for? What is the value of when 0? 3 To get the -term b itself on the left side, subtract from each side. 3 Simplif. 3 Divide each side b since is being multiplied b on the left side. This isolates. 3 (0) 3 7 Simplif b dividing each term b. Notice, this changes the sign of each term. To find the value of when 0, substitute 0 in for. Simplif b multipling first, then subtracting. When ou rewrite literal equations, ou ma have to divide b a variable or variable epression. When ou do so in this lesson, assume that the variable or variable epression is not equal to zero because division b zero is not defined. Solve the equation ab bc cd for b. b(a c) cd Since b is a factor of each term on the left side, it can be factored out using the Distributive Propert. b(a c) a c a cd c To get b b itself, divide each side b a c since b is being multiplied b a c. Remember a c 0. b a cd c Simplif. Solve each equation for. Then find the value of for each value of.. ;, 0,. 6 ;,, ;, 0, ; 7; ; 3; 3 ; ; 8; ; 3; ; 3; 7;. 8;,, 0. 3 ; 0,, ;, 0, 8 ; ; 3 ; ; 7. 3( ) ;, 0, 8. 3 ; 3 ; 3; 3 3 ; 6 8 ; 8 ; ;, 0, 9. 3;,, 7 3 ; 3 ; 3 ; ; ; ; ; 6; 3 ; 7; ; ; 9

29 - Reteaching (continued) Literal Equations and Formulas A formula is an equation that states a relationship among quantities. Formulas are special tpes of literal equations. Some common formulas are shown below. Notice that some of the formulas use the same variables, but the definitions of the variables are different. For instance, r is the radius in the area and circumference of a circle and the rate in the distance formula. Formula Name Perimeter of a rectangle Circumference of a circle Area of a rectangle Formula P l w C πr A lw Area of a triangle A bh Area of a circle A πr Distance traveled d rt Each of the formulas can be solved for an of the other unknowns in the equation to produce a new formula. For eample, r C π is a formula for the radius of a circle in terms of its circumference. What is the length of a rectangle with width cm and area 6 cm? A lw Formula for the area of a rectangle. A w lw w Since ou are tring to get l b itself, divide each side b w. l w A l 6 l 6 cm Simplif. Substitute 6 for A and for w. Simplif. Solve each problem. Round to the nearest tenth, if necessar. Use 3. for π. 0. A triangle has base 6 cm and area cm. What is the height of the triangle? cm. What is the radius of a circle with circumference 6 in.? about 8.9 in.. A rectangle has perimeter 80 m and length 7 m. What is the width? 3 m 3. What is the length of a rectangle with area 0 ft and width ft? 33. ft. What is the radius of a circle with circumference 7 in.? about.3 in. 0

30 -6 Reteaching Ratios, Rates, and Conversions A unit rate is a rate with denominator. For eample, can be used to compare quantities and convert units. in. ft is a unit rate. Unit rates Which is greater, 7 inches or 6 feet? It is helpful to convert to the same units. Conversion factors, a ratio of two equivalent measures in different units, are used to do conversions. Multipl the original quantit b the conversion factor(s) so that units cancel out, leaving ou with the desired units. 6 ft 3 in ft 7 in. Since 7 in. is less than 7 in., 7 in. is greater than 6 ft. Rates, which involve two different units, can also be converted. Since rates involve two different units, ou must multipl b two conversion factors to change both of the units. Jared s car gets 6 mi per gal. What is his fuel efficienc in kilometers per liter? You need to convert miles to kilometers and gallons to liters. This will involve multipling b two conversion factors..6 km mi There are.6 km in mi. The conversion factor is either mi or.6 km..6 km Since miles is in the numerator of the original quantit, use mi as the conversion factor so that miles will cancel. 6 mi.6 km 3 gal mi There are 3.8 L in gal. The conversion factor is either 3.8 L gal or gal 3.8 L. Since gallons is in the denominator of the original quantit, use gal 3.8 L as the conversion factor so that gallons will cancel. 6 mi.6 km 3 gal mi 3 gal 3.8 L < 0.9 km L Jared s vehicle gets 0.9 kilometers per liter. 9

31 -6 Reteaching (continued) Ratios, Rates, and Conversions Convert the given amount to the given unit.. hours; minutes. 000 cm; km 3. ft; d 70 min 0.0 km d. 3 cups; gallons. 30 m; cm 6. lbs; kilograms gal 3000 cm 6.8 kg 7. in.; cm 8. 0 miles; km 9. ft; in cm 6.09 km 300 in. 0. Serra rode mi in. hr. Phaelon rode 38 mi in 3. h. Justice rode mi in. hr. Who had the fastest average speed? Phaelon. Mr. Hintz purchased gallons of drinking water for his famil for $.8. He knows that this should last for weeks. What is the average cost per da for drinking water for the famil? $.0/da. The price for a particular herb is 9 cents for 6 ounces. What is the price of the herb in dollars per pound? $.3/lb Cop and complete each statement. 3. mi/h ft/s. 7 g/s kg/min. 0 cents/min $/h 66 ft/s 0. kg/min $ 30/h 6. m/h cm/s 7. km/min mi/h 8. 6 gal/min qt/h 0.6 cm/s 8.9 mi/h 0 qt/h 9. Writing Describe the conversion factor ou would use to convert feet to miles. How do ou determine which units to place in the numerator and the denominator? mile is the numerator and 80 ft is the denominator; ft should cancel out so ft should be in the denominator 0. Writing Describe a unit rate. How do ou determine the unit rate if the rate is not given as a unit rate. Illustrate using an eample. A unit rate is a rate with a denominator of ; divide both the numerator and denominator b the denominator; lbs $3 3 lb.6 lb $3 3 $ 60

32 -7 Reteaching Solving Proportions A proportion is an equation that states that two ratios are equal. If a quantit in a proportion is unknown, ou can solve a proportion to find the unknown quantit as shown below. What is the solution of 3? There are two methods for solving proportions using the Multiplication Propert of Equalit and the Cross Products Propert. ) The multiplication Propert of Equalit sas that ou can multipl both sides of an equation b the same number without changing the value. 3 Q 3 R Q R To isolate, multipl each side b. Simplif. 0. Divide b. ) The Cross Products Propert sas that ou can multipl diagonall across the proportion and these products are equal. 3 ()() (3)() Multipl diagonall across the proportion. 0. Multipl. To isolate, divide each side b. Simplif. Real world situations can be modeled using proportions. A baker can make 6 dozen donuts ever minutes. How man donuts can the baker make in hours? A proportion can be used to answer this question. It is ke for ou to set up the proportion with matching units in both numerators and both denominators. For this problem, ou know that hours is 0 minutes and 6 dozen is 7 donuts. Correct: 7 donuts min donuts 0 min Incorrect: 7 donuts 0 min min donuts 69

33 -7 Reteaching (continued) Solving Proportions This proportion can be solved using the Multiplication Propert of Equalit or the Cross Products Propert. Solve this proportion using the cross products. 7 donuts min donuts 0 min (7)(0) Cross Products Propert 860 Multipl. 860 Divide each side b..3 Simplif. Since ou cannot make 0.3 donuts, the correct answer is donuts. Solve each proportion using the Multiplication Propert of Equalit.. 3 n 7. z t a 3. n d 8 7 Solve each proportion using the Cross Products Propert b 8 8. m z v f h A cookie recipe calls for a half cup of chocolate chips per 3 dozen cookies. How man cups of chocolate chips should be used for 0 dozen cookies? 3 Solve each proportion using an method

34 -8 Reteaching Proportions and Similar Figures In similar figures, the measures of corresponding angles are equal, and the ratios of corresponding side lengths are equal. It is important to be able to identif the corresponding parts in similar figures. B E Since /A > /D, /B > /E, and /C > /F, AB DE BC EF, AB DE AC DF. This fact can help ou to find missing lengths. A C D F What is the missing length in the similar figures? First, determine which sides correspond. The side with length corresponds to the side with length 6. The side with length corresponds to the side with length. These can be set into a proportion. 6 (6)() ()() Write a proportion using corresponding lengths. Cross Products Propert Multipl. Divide each side b 6 and simplif. 6 The figures in each pair are similar. Identif the corresponding sides and angles.. F G. B C N 0 m m Q 60 m 6. m A D E H M 8 m O P 7 m R AB and EF, BC and FG, CD and GH, HE and DA, la and le, lb and lf, lc and lg, ld and lh MN and PQ, NO and QR, OM and RP, lm and lp, ln and lq, lo and lr 79

35 -8 Reteaching (continued) Proportions and Similar Figures The figures in each pair are similar. Find the missing length m. cm 8 m m m about. cm cm 9 cm. ft 0 ft ft 6. 3 in.. ft 7 in. 3 in. 8 in. A map shows the distance between two towns is 3. inches where the scale on the map is 0. in. : mi. What is the actual distance between the towns? map distance Map scale: actual distance If ou let be the actual distance between the towns, ou can set up and solve a the proportion to answer the question. 0. in. 3. in. mi mi The towns are 70 miles apart. The scale of a map is. in. : 0 mi. Find the actual distance corresponding to each map distance in. 8.. in in mi 3 mi mi 0. The blueprints of an octagonal shaped hot tub are drawn with a in. : ft scale. In the drawing the sides are 3. inches long. What is the perimeter of the hot tub? 0 ft 80

36 -9 Reteaching Percents Percents compare whole quantities, represented b 00%, and parts of the whole. What percent of 90 is 7? There are two was presented for finding percents. ) You can use the percent proportion a b p 00. The percent is represented b p 00. The base, b, is the whole quantit and must be the denominator of the other fraction in the proportion. The part of the quantit is represented b a p 00 7(00) (90)(p) Cross Products Propert p Multipl. Substitute given values into the percent proportion. Since ou are looking for percent, p is the unknown. 30 p Divide each side b 90 and simplif. 7 is 30% of 90. ) The other wa to find percents is to use the percent equation. The percent equation is a p% 3 b, where p is the percent, a is the part, and b is the base. 7 p% 3 90 Substitute 7 for a and 90 for b. 0.3 p% Divide each side b % p% Write the decimal as a percent. 7 is 30% of 90. Find each percent.. What percent of is 0? 0%. What percent of is 3? 0% 3. What percent of is 8? 7%. What percent of 0 is 7? 0% 7% of 96 is what number? In this problem ou are given the percent p and the whole quantit (base) b. a p% 3 b Write the percent equation. a 7% Substitute 7 for p and 96 for b. Multipl. 89

37 -9 Reteaching (continued) Percents 8% of what number is? You are given the percent p and the partial quantit a. You are looking for the base b. a p% 3 b Write the percent equation. 8% 3 b Substitute 8 for p and for a b Write 8% as a decimal, b Divide each side b 0.8. Find each part.. What is 3% of 0? What is 78% of 30? 0. Find each base. 7. % of what number is 90? % of what number is 3? 0 s involving simple interest can be solved using the formula I Prt, where I is the interest, P is the principal, r is the annual interest rate written as a decimal, and t is the time in ears. You deposited $00 in a savings account that earns a simple interest rate of.8% per ear. You want to keep the mone in the account for 3 ears. How much interest will ou earn? I Prt Simple Interest Formula I (00)(.8%)(3) Substitute 00 for P,.8% for r, and 3 for t. I 8.8 You will earn $8.80 in interest. Multipl. 9. If ou deposit $,000 in a savings account that earns simple interest at a rate of 3.% per ear, how much interest will ou have earned after ears? $ 9 0. If ou deposit $00 in a savings account that earns simple interest at a rate of.% per ear, how much interest will ou have earned after 0 ears? $.0 90

38 -0 Reteaching Change Epressed as a Percent A percent change occurs when the original amount changes and the change is epressed as a percent of the original amount. There are two possibilities for percent change: percent increase or perent decrease. The following formula can be used to find percents of increase/decrease. percent change amount of increase or decrease original amount In its first ear, membership of the communit involvement club was 3 members. The second and third ears there were 8 members and 3 members respectivel. Determine the percent change in membership each ear. From the first to the second ear, the membership went down from 3 to 8 members, representing a percent decrease. The amount of decrease can be found b subtracting the new amount from the original amount. percent change original amount new amount original amount Percent Change Formula for percent decrease. Substitute 3 for the original number and 8 for the new number Subtract. Then divide. Membership decreased b.% from the first ear to the second ear. From the second to the third ear, the membership increased from 8 to 3 members, representing a percent increase. The amount of increase can be found b subtracting the original amount from the new amount. percent change original amount new amount original amount Percent Change Formula for percent increase. Substitute 8 for the original number and 3 for the new number. 7 3 < 0. Subtract. Then divide. Membership increased b about % from the second ear to the third ear. Tell whether each percent change is an increase or decrease. Then find the percent change. Round to the nearest percent.. Original amount:. Original amount: 7 3. Original amount: New amount: New amount: 0 New amount: increase; 80% decrease; % decrease; %

39 -0 Reteaching (continued) Change Epressed as a Percent Errors can occur when making measurements or estimations. Percents can be used to compare estimated or measured values to eact values. This is called relative error. Relative error can be determined with the following formula comparing the estimated value and the actual value. Percent error u measured or estimated value actual value u actual value Mrs. Desoto estimated that her class would earn an average of $6 per person for the fundraiser. When the mone was counted after the fundraiser ended, each student had raised an average of $38 per person. What is the percent error? There are two values given in this situation. The estimated value is $6 per person. The actual value that each person raised was $38. u measured or estimated value actual value u Percent error actual value u 6 38 u 38 Percent Error Formula u u 38 Subtract. 38 u u < 0.09 Divide. There was a 9% error in her estimation. Find the percent error in each estimation. Round to the nearest percent. Substitute 6 for the estimated value and 38 for the actual value.. You estimate that our bab sister weighs lbs. She is actuall 6 lbs. %. You estimate that the bridge is 60 ft long. The bridge is actuall 3 ft long. 3% 6. You estimate the rope length to be 80 ft. The rope measures 7 ft long. % 7. A carpenter estimates the roof to be 37 ft. The rectangular roof measures 8 feet wide b feet long. What is the percent error? % 00

40 3- Reteaching Inequalities and Their Graphs You use the following smbols for inequalities.. is greater than $ is greater than or equal to, is less than # is less than or equal to What inequalit represents plus a number is less than 0? plus a number is less than 0, 0 The inequalit,0 represents the phrase. Write an inequalit that represents each verbal epression.. p is greater than or equal to p L. a is less than or equal to a K 3. times d is less than 0 d R 0. r divided b is greater than 0 r S 0 Is a solution of 3t 0 $? 3t 0 $? Original inequalit 3() 0 $? Substitute for t. 6 0 $ Simplif. is not a solution. Determine whether each number is a solution of the given inequalit.. b 7. 3 a. b. c. 8 no no es 6. (m ),6 a. 6 b. c. es no no 7. 8 h # 8 a. 6 b. 8 c. 0 es es no 9

41 3- Reteaching (continued) Inequalities and Their Graphs When graphing an inequalit on a number line, an open circle means the number is not included in the inequalit. A closed circle means the number is included in the inequalit. What is the graph of w $? Since w is greater than or equal to, place a closed circle at. Draw a dark line with an arrow to the right of the closed circle to show the numbers greater than. 0 Graph each inequalit. 8. # 0 9. p a $ What inequalit represents the graph? The circle is open so is not included in the inequalit. The dark line and arrow are to the left indicating less than. The graph represents is less than or,. Write an inequalit for each graph R L K S 0

42 3- Reteaching Solving Inequalities Using Addition or Subtraction You can add the same number to each side of an equation. You can also add the same number to each side of an inequalit. What are the solutions of b.? Graph and check the solutions. b. Original inequalit. b. Add to each side. b. Simplif. To graph b., place an open circle at and shade to the right. 0 To check the endpoint of b., make sure that is the solution of the related equation b. b 0 Then check to see if a number greater than is a solution of the inequalit. is greater than. b.?.. Solve each inequalit. Graph and check our solutions.. m $0 m L. t, t R # K 7. d 9 $ d L w 7. 3 w S a,7 a R Writing Eplain how ou would solve t #. add to both sides 8. Anita is baking dinner rolls and pumpkin bread. She needs cups of flour for the rolls. She needs at least 7 cups of flour left for the pumpkin bread. Write and solve an inequalit to determine how much flour Anita needs before she starts baking. c L7; at least cups 9

43 3- Reteaching (continued) Solving Inequalities Using Addition or Subtraction You can subtract the same number from each side of an equation. You can also subtract the same number from each side of an inequalit. What are the solutions of h 7 #? Graph and check the solutions. h 7 # Original inequalit. h 7 7 # 7 Subtract 7 from each side. h #3 Simplif. To graph h #3, place a closed circle at 3 and shade to the left. 0 To check the endpoint of h #3, make sure that 3 is the solution of the related equation h 7. h Then check to see if a number less than 3 is a solution of the inequalit. is less than 3. h 7 #? 7 # 3 # Solve each inequalit. Graph and check our solutions. 9. s 7 $ s L 0. p 3, p R b # b K 9. n $ 8 n L v 8. v S 30. k 6, 6 k R A boat can hold up to 000 pounds. Two friends get in the boat. Together the weigh 8 pounds. Write and solve an inequalit to determine how much more weight can be added to the boat. w 8 K000; up to 7 pounds 0