Simplifying Expressions

Transcription

1 UNIT 1 Math 621 Simplifying Expressions Description: This unit focuses on using quantities to model and analyze situations, interpret expressions, and create expressions to describe situations. Students will work in one-variable and multivariable settings, attain fluency with such algebraic manipulations as use of the distributive property, simple factoring, and connecting the structure of expressions to the contextual meanings of those expressions. Broadly, the unit will help students with the transition into using the tools of algebra to model and explore scenarios. 1.1: Order of Arithmetic Operations (PEMDAS) 1.2: Linear Combinations and Writing Math Expressions 1.3: Arithmetic Properties 1.4: Greatest Common Factors (GCF) and simple factoring 1.5: Evaluating expressions 1.6: Simplifying expressions & combining like terms 1.7: Review Unit 1 TEST

2 RESOURCES 1.1 PEMDAS Khan Academy PEMDAS: 1.2: Linear Combination and Writing Math Expressions How to Write Expressions from Variables: 1.3: Properties (associative, commutative, distributive, additive/mult identities, additive/mult inverses) Arithmetic Properties: 1.4: GCF and simple factoring GCF: 1.5: Evaluating expressions (substitution) - (more practice on PEMDAS in the context of substitution) Evaluating Expressions with Multiple Variables: 1.6: Simplifying expressions & combining like terms Simplifying using Distributive Property and Combining Like Terms:

3 Section 1.1 Notes The Order of Operations A set of guidelines used to simplify mathematical expressions. When simplifying an expression, the order is perform all operations inside any parenthesis first, followed by evaluating all exponents. Third, do all multiplication and division at the same time, from left to right. Lastly, do all addition and subtraction at the same time, from left to right. Some people refer to the order of operations as PEMDAS. The order is as follows: Parenthesis Do anything in parenthesis first. Exponents Next, all powers (exponents) need to be evaluated. Multiplication/Division Multiplication and division must be done at the same time, from left to right, because they are inverses of each other. Addition/Subtraction Addition and subtraction are also done together, from left to right. Example A Simplify !2 " (5 "1) Solution: Parenthesis: ! 3" 4 Example B Simplify 9! " 3! 7 Exponents: 4 + 6! 3" 4 Multiplication: 4 +18! 4 Add/Subtract: 18 Solution: Think of everything in the numerator as if it were in its own set of parenthesis as well as everything in the denominator. The problem can be rewritten as (9! ) (2 2 " 3! 7) When there are multiple operations in a set of parenthesis, use the Order of Operations within each set. (9! ) (2 2 " 3! 7) = (9! 2 +13) (4 " 3! 7) = (7 +13) (12! 7) = 20 5 = 4 page 1

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5 Section 1.1 PEMDAS classwork Name Evaluate these expressions using correct order of operations. Show your steps. No calculators (2 + 3) (8 3) * * 9 3(6 1) (5 2) [5 + (4 1) 2 ] [- 1(- 2 1)] 2 page 3

6 [8 3(4 2)] x + 5[6 (2x + 3)] Make the following expressions equal to 35 by placing parenthesis = = 35 page 4

7 Section 1.2 Algebraic Expressions Name Match each English phrase or sentence with its algebraic translation. 1. Five less than twice a number. 2. Five more than twice a number. 3. Five is less than twice a number. 4. Five more than twice a number is less than ten. 5. Five less than twice a number is greater than ten times the number. 6. Five less than twice times a number is less than the number. (a) 2n + 5 (b) 5 > 2n (c) 5 < 2n (d) 5 2n (e) 2n 5 (f) 2n + 5 > 10 (g) 2n (h) 2n 5 > 10n (i) 2n 5 < n (j) 2n + 5 < 10 (k) 2n 5 > 10 (l) 2n 5 < 10 Translate into words. 7. > Translate into words as illustrated in the example. example: n + 4 Four more than a number. 11. x w n 14. 5n y < 8 page 5

8 Section 1.2 Notes Algebraic Expressions (part 2) A linear combination is an expression in the form. Example 1: In a professional hockey game, a win is worth two points, a tie is worth one point and a loss is worth zero points. Write an expression to express the total number points scored given the number of wins (W), the number of ties (T) and the number of losses (L). Example 2: Ivan bought P pounds of peaches at $2.29 per pound and G pounds of grapes at $3.79 per pound. a. Write an expression that gives the amount Ivan paid for the peaches and the grapes. b. Suppose he bought 2 pounds of peaches and 3 pounds of grapes. How much did he spend in total? c. Suppose he spent $15.24 in total. If he only bought one pound of grapes, how many pounds of peaches did he buy? page 6

9 Section 1.2 Practice 1. The expression 37P + 32R is called a of P and R. 2. Suppose you bought D DVD s and B Books. If each DVD cost $14.95 and each book you bought off the specials table cost $6.95, write an expression that tells how much you spent. 3. On a quiz show, 20 points are given for correct answers to regular questions and 50 points are given for correct answers in the bonus round. Let R represent the number of regular questions answered correctly and B represent the number of bonus questions answered correctly. a. Write an expression that gives the total number of points earned. b. Suppose that a contestant earned 650 points. Write an equation relating R, B, and the number of points earned. c. Give three different possible solutions to the equation you wrote in part b. Review: Translate into an algebraic expression or sentence. 1. p less than y 2. p is less than y 3. Five more than three times a number page 7

10 Choose the correct answer for each question. (a) x + y (b) x y (c) xy (d) x y (e) y x 4. You give a friend y dollars. You had x dollars. How much do you have left? 5. You drove x miles in y hours. What was your rate? 6. You buy x pencils at y cents each. What is the total cost? 7. Sara is y years old. Her sister is x years older. How old is her sister? page 8

11 Section 1.3 Notes Arithmetic Properties The properties of arithmetic enable us to solve mathematical equations. Notice that these properties hold for addition and multiplication. Name Form Example Commutative Property of Addition a + b = b + a Commutative Property of Multiplication ab = ba Associative Property of Addition a + (b + c) = (a + b) + c Associative Property of Multiplication a(bc) = (ab)c Additive Identity a + 0 = a Multiplicative Identity a 1 = a Additive Inverse a + (!a) = 0 Multiplicative Inverse a! 1 a = 1 Distributive Property of Multiplication over Addition a(b + c) = ab + ac From the Identity Property, we can say that 0 is the additive identity and 1 is the multiplicative identity. Similarly, from the Inverse Property,!a is the additive inverse of a and 1 a is the multiplicative inverse of a because they both equal the identity for their respective operations. page 9

12 Example Identify the property used in the equations below. a) 3! 4!5 = 5! 4! 3 b) 5! 1 5 = 1 c) 6!(7!8) = (6! 7)!8 d) 6 + (! 6) = 0 e) 6 + (2 + 3) = (6 + 2) + 3 f) = g) 2(4x! 3) = 8x! 6 h) = 4 page 10

13 Section 1.3 Arithmetic Properties Practice Sheet #1 Name Match each example with the property that is illustrated. (a) Additive Identity (b) Additive Inverse (c) Associative Property of Addition (d) Associative Property of Multiplication (e) Commutative Property of Addition (f) Commutative Property of Multiplication (g) Distributive Property of Mult over Addn (h) Multiplicative Identity (i) Multiplicative Inverse 1. 2(3 + 4x) = 6 + 8x (3 + 4) = (2 + 3) (- 3) = 0 4. ( 5! 3)! 7 = 5!( 3! 7) 5. 10!0 = (2 + 5)! 4 = 2! 4 + 5! = ! 5 3 = 1 9. (3 + 7) + (2 + 1) = (2 + 1) + (3 + 7) 10. (7!2)! 4 = (2! 7)! = ! 1 12 = Which statement illustrates the associative property of multiplication? (a) ( 3a)b = b( 3a) (b) ( 3a)b = 3ab (c) ( 3a)b = 3( ab) (d) ( 3a)b = ( a! 3)b page 11

14 14. Which statement illustrates the additive identity? (a) 2 2 = 0 (b) =2 (c) 2!1 = 2 (d) 2! 1 2 = 1 " 15. Use the distributive property to rewrite 16 x! 1 % # $ 4 & ' without parentheses. (a) 16x - 20 (b) 16x (c) 16x - 16 (d) 16x - 4 page 12

15 1.3 Properties of Real Numbers Practice Sheet #2 Name Match each example with the property that is illustrated. (a) Additive Identity (b) Additive Inverse (c) Associative Property of Addition (d) Associative Property of Multiplication (e) Commutative Property of Addition (f) Commutative Property of Multiplication (g) Distributive Property of Mult over Addn (h) Multiplicative Identity (i) Multiplicative Inverse 1. m + 3 = 3 + m 10. m!a!t!h = h!a! m!t 2. w!1 = w 11. (!y) + y = 0 3. a + (b + c) = (a + b) + c 12. ab + xy = ba + yx 4. ( 2! x)! y = ( x!2)! y n!n = x(x + 3) = 2x 2 + 6x 14. (2 + m) + 3 = 3+ (2 + m) 6. k + (!k) = a!( j!e) = ( a! j)!e 7. (a + b) + c = a + (b + c) 16. ( 2!a)!b = 2!( a!b) 8. u + 0 = u 17. 2x + 6 = 2(x + 3) 9. r! 1 r = h = h page 13

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17 1.3 Properties of Real Numbers Classwork #3 Name Notice that these properties hold for addition and multiplication. Commutative Property of Addition/Multiplication Associative Property of Addition/Multiplication Distributive Property of Multiplication over Addition Additive Inverse Multiplicative Inverse Additive Identity Multiplicative Identity Problems. All variables represent real numbers. 1. Use the Commutative Property of Addition to rewrite the expression 5x Use the Commutative Property of Multiplication to rewrite the expression 2! Use the Associative Property of Addition to rewrite the expression (7x + 4) Use the Associative Property of Multiplication to rewrite the expression 7!(4!8) 5. Use the Distributive Property to rewrite the expression 6(x + 4). 6. Use the Distributive Property to rewrite the expression 5x+10. Identify which property is illustrated in each statement below. 7. x(yz) = x(zy) 8. 2(x + y) = 2x + 2y 9. (x + y) + z = (y + x) + z 10. (x + y) + z = x + (y + z) 11. 2[x (a + d)] = (2x) (a + d) ! 1 47 = a + 3b = 3(a + b) 14. x + (-x) = 0 page 15

18 15. Show using examples that Subtraction and Division are NOT commutative or associative operations. 16. Does x( y! z)!=!xy! xz? What property is being used here? page 16

19 Section 1.4 GCF and Simple Factoring Name: GCF: The Greatest Common Factor of a set of numbers is Method for finding the GCF: Example: Find the GCF of 20 and 45 a. List the factors of each number. 20: b. Identify the common factors. c. Choose the greatest of those. 45: Sometimes the GCF includes a variable. For example, let s look at 4x 2 and 16x 5. 4x 2 : 16x 5 : Try it! Find the GCF of the following sets of numbers , , 20, a 3, 15a 6, 10a 4. 3k 4, 8k, 5k 5 page 17

20 How to Factor Out the GCF from an expression: Now we are going to rewrite expressions so that they are a multiplication of the GCF and the rest of the expression. Example: 4x+20 a. Identify the GCF. b. Divide each term in the expression by the GCF. How can you check your final expression is equivalent to the original? Note: If there is no GCF, that means the expression is already as simplified as possible and is called prime. Let s try it together! Factor out the GCF from each expression. Write PRIME if there is no GCF x 2. 12y x d + 45d 4. 13a c + c c n 30n+ 42 page 18

21 Now try it on your own. Factor out the GCF from each expression. Write PRIME if there is no GCF. 7. 5m 2!10m p p p 9. 18x 2!50x z + 50z 75z g + 3g k 30 CHALLENGE!!! Push yourself! cd 3c + 4c rs 108r s x x + x x y y y y page 19

22 Section 1.5 Notes Evaluating Expressions Name: You have probably seen letters in a mathematical expression, such as 3x + 8. These letters, also called variables, represent an unknown number. One of the goals of algebra is to solve various equations for a variable. To evaluate an expression or equation, we would need to substitute in a given value for the variable and test it. In order for the given value to be true for an equation, the two sides of the equation must simplify to the same number. Example A: Evaluate 2x 2 9 for x = 3. Example B: Determine if x = 5 is a solution to 3x 11 = 14. Example C: Sometimes you are given a formula to use. For example, the formula for the area of a circle is A =!r 2, where r is the radius in feet. If the radius is 4ft, find the area. Vocabulary Check Variable- A letter used to represent an unknown value. Expression- A group of variables, numbers, and operators. Equation- Two expressions joined by an equal sign. Solution- A numeric value that makes an equation true page 20

23 1.5 Classwork Below are some formulas that we will use for practice today. You do NOT need to memorize these. Area of a Circle with radius = r. A =!r 2 Surface area of a rectangular prism with length = l, width = w and height = h. SA = 2wl + 2wh + 2lh Volume of a cone with radius = r and height = h. V = 1 3!r2 h r Children s Dose of some Medications given the adult dose = A and the child s age = g (years).! g $ C = # &' A " g +12 % Volume of a Sphere with radius = r V = 4 3!r3 Surface Area of a Cylinder with radius = r and height = h. SA = 2!r 2 + 2!rh Use the appropriate formula from the list above to find the values requested below. 1. Find the surface area of a rectangular prism 2. Find the volume of a cone with radius = 3 cm with length = 10 in, width = 4 in, & and height = 12 cm. height = 5 in. 3. Find the Area of a circle with radius = 5 in. 4. Find the appropriate dose of a medication for a 6 year old child if the Adult dose is 120 mg. page 21

24 5. Find the surface area of a cylinder with radius = 5 cm and height = 20 cm. 6. Find the volume of a sphere with radius = 6 in. 7. Here s another useful formula. Suppose you stand at the top of a tall building and you drop something (like a penny) off the roof. The distance the object has fallen depends upon how long it has been falling. Let t stand for time (in seconds). If we measure the distance in feet, the formula is: d = 1 2 (32)t 2. If we measure in meters, it is: d = 1 2 (9.8)t Note: The general formula is d = 1 2 gt 2 8. Suppose you stand at the top of a very tall building and drop a penny off the top. How many feet will it have traveled after 2 second? Suppose you drop a penny off the same building. How many meters will it have travelled in 2.5 seconds? where g stands for gravitational constant. The gravitation constant for feet = 32 ft/sec and the gravitation constant for meters = 9.8 m/sec. for objects near the surface of the Earth. The numbers are different near the surface of the moon or planets other than Earth. Evaluate each variable expression when a = 4 and b = (a + b) b 2 + 2ab b! a + 7 a 2 + b page 22

25 Section 1.6 Notes Simplifying Expressions Corey has a bowl of fruit that consists of 5 apples, 4 oranges, and 3 limes. Katelyn went to the farmer s market and picked up 2 apples, 5 limes, and an orange. How many apples, oranges, and limes do Corey and Katelyn have combined? Combining like terms is much like grouping together different fruits, like apples and oranges. Every algebraic expression has at least one term. A term is a number or is the product of a number and a variable. Terms are separated by addition and subtraction signs. A constant is a term that has no variable. Given: 3x + 7 3x and 7 are terms (7 is also called a constant) Algebraic expressions often have like terms that can be combined. Like terms are terms that have the same variable. All constants are like terms. The coefficient, the number multiplied by a variable in a term, does not need to be the same in order for the terms to be like terms. Example 1: Match pairs of like terms: 6x 2y 3m x 9 3m 5 14y In order to simplify an algebraic expression you must combine all like terms. When combining like terms you must remember that the operation in front of the term (addition or subtraction) must stay attached to the term. Rewrite the expression by grouping like terms together before adding or subtracting the coefficients to simplify. Example 2: Simplify each algebraic expression by combining like terms. a. 4x + 3x + 5x b. 8y y + 1 c. 9m + 10p 3p 2m + 4m d. 4d d 12 Simplify each algebraic expression by combining like terms. 1. 4y + 2y + 3y 4. 5x + 9y + 3x 2y 7. 22u 6u + 4t + 4t 8u 2. 8x + x 5x 5. 14p p 8. x + y + x + x + 2y x 3. 10m 4m + 2m 3m d 3d n + 3n 2 13 page 23

26 1.6 Classwork Combining Like Terms Name Combine like terms within each expression 1. 8x 10 5x a 7b + 5a 20b a 2 + 5! 3a 2 + 5a 3! 3a x! 3y + 8x + 5y 5. 3a 2! 4b + 2b 2! 3a + b 2 + 4b + 3a 6. 12x! 5! 7x 2 + 3x + 6 Find the GCF of the following expressions and use the Distributive Property to simplify each one. 7. 5x a a 2! x! 25y a + 8b a 2!14a + 21 We can also use the Distributive Property and GCF to pull out common variables from an expression. Find the GCF and use the Distributive Property to simplify the following expressions x 2 + 4x 14. a 3! 5a 2 + a 15. 5a 2 +10a 16. 3y 3 + 9y a 2 + 8ab m 4! 40m m 2 page 24

27 Unit 1 Review & Study Guide Name Lesson 1.1: You should know the order of arithmetic operations (PEMDAS) and be able to use them to simplify arithmetic expressions. Simplify the following expressions using the correct order of operations (2!5) ! 2(7 + 9) + 5 " (( 17 6)!2) ( 11 4) + ( ) ! !( ) 3 3 Lesson 1.2: You should be able to translate English sentences into mathematics sentences and to use mathematical expressions to represent real- life situations. Translate these mathematical symbols into words 7. > Write an algebraic expression or equation for each situation. 11. A number is 6 less than twice another number. 12. Four greater than a number. 13. A number is greater than 4 times another number. 14. Liz bought food for a party. Each pizza cost $7.99, each bottle of soda was $2.99 and each bag of chips was $1.49. How much did she spend in total? page 25

28 15. Hannah bought G bags of Gold Fish at $2.29 per bag and C bags of Chex Mix at $1.79 per bag. a. Write an expression that gives the amount Hannah paid for all of the food. b. Suppose she bought 5 bags of Gold Fish and 3 bags of Chex Mix. How much did she spend in total? c. Suppose she spent $13.03 in total. If he only bought one bag of Gold Fish, how many bags of Chex Mix did she buy? Lesson 1.3: You should know the properties of arithmetic and be able to identify which property is illustrated given a mathematical sentence. You should also be able to give examples of properties. Match the example column with the proper property given on the left column a. Commutative Property of Addition b. Associative Property of Multiplication c. Distributive Property of Multiplication over Addition d. Additive Inverse e. Multiplicative Inverse 16. m! s = s! m 17. a + 0 = a 18. a! 1 a = x(m + n) = xm + xn 20. p + - p = 0 f. Additive Identity 21. ( m! p)! s = m!( p! s) g. Multiplicative Identity h. Commutative Property of Multiplication i. Associative Property of Addition 22. xy + 2z = 2z + xy 23. t!1 = t 24. (m + p) + s = m + (p + s) 25. k!5 = 5k 26. 7x + 7 j = 7(x + j) 27. a + (b + c) = a + (c + b) page 26

29 Lesson 1.4: You should be able to factor linear expressions using the GCF Find the GCF and use the Distributive Property to simplify the following expressions x + 3y 29. 6x 2! c! x! 2y +10z 32. x 2! 9x 33. 2m 3 + 6m m x + 8y! 60z 35. 3x 2 + 6x Lesson 1.5 You should be able to evaluate expressions & equations 36. If x = 3: 3! x 5! x 37. Solve for V given that V = pr 2 h and p = 3.14, r = 4 and h = 7 Evaluate the following expressions given x = 3, y = 5, and z = z(x 2! 6) +1 y 39. 3x + 4z 5( y + xz) 40. The formula A = 1 P i a gives us the area A for a regular polygon. The variable a represents the 2 length of the apothem and P represents the perimeter. If the perimeter is 36 cm and the apothem is about 5 1 cm, find the area for the polygon. 5 page 27

30 Lesson 1.6 You should be able to simplify algebraic equations by combining like terms Simplify the following expressions as much as possible p! 3p 42. 3x! 2! 8x 43. 5u 2! 3u! 7u 2 + 9u (2x + 4)!15x! (x + 2)! (x! 3) 46. 5x + 3(x! 2) 47. 3x + x(5! 2x) x + 6! 2(3x + 8) (12x + 20)! 1 5 (30!15x) (2x! 9)! 0.6(4x + 5) page 28

31 Unit 1 Review & Study Guide Answers (2!5) 2 = = = 109 ( 11 4) + ( ) = 7 + (15! 5) = = ! 2(7 + 9) + 5 "2 2 = 35! 2(16) + 5 " 4 = 35! = ! 3+1 = ! 3+1 = = (( 17 6)!2) ( ) 3 3 ( ) ! = 10! 2 4 = 10!("2) 3 3 = 10!( " 8) 3 = "80 " 3 = "83 7. Greater than 8. not equal 9. less than or equal 10. approximately 11. n = 2x n n > 4x p s c 15a. 2.29G C 15b, 5(2.29) + 3(1.79) = $ c C = 13.03; 1.79C = 10.74; C = 6 answer: 6 bags of Chex Mix 16. h 17. f 18. e 19. c 20. d 21. b 22. a 23. g 24. i 25. h 26. c 27. a 28. 3(x + y) 29. 6(x 2! 7) 30. 7(c 2) 31. 2(x y + 5z0 32. x(x 9) 33. 2m(m 2 + 3m +11) 34. 4(3x + 2y 15 z) 35. 3x(x + 2) 36. 3! 3 5! 3 = V = 3.14 (16) (7) = !2(9! 6) +1 = 5!6 +1 = =!5 5 5 =!1!2(3) (3) + 4(!2)! 5(5 + (3"!2)) = 9 +! 8! 5(5 +! 6) = 1! 5(!1) = 1+ 5 = A = 1 (36)(5.2) = 93.6 cm p x 43.!2u 2 + 6u x 45. 6x x 6 47.!2x 2 + 8x x x x 6.6 page 29

32 Unit 1 Test Review #2 Name: Evaluate these expressions using the correct order of operations. Show your steps. 1. (6 + 3) (9! 2 2 ) 2. 9!10 5 " ! (6!8) ! " ! [ (2 " 4)] [- 3(- 8 2)] 2 Add parenthesis to make each equation true !1" 4 +1 = 9 5 2! = 10 Insert the proper operation signs (+,, x, ) and grouping symbols, when needed, to make each sentence true = = 9 page 30

33 Match each example with the property that is illustrated. (a) Additive Identity (b) Additive Inverse (c) Associative Property of Addition (d) Associative Property of Multiplication (e) Commutative Property of Addition (f) Commutative Property of Multiplication (g) Distributive Property of Mult over Addn (h) Multiplicative Identity (i) Multiplicative Inverse 11. 3x + 2y = 2y + 3x! 17. 4! 3 = 3! x + 0 = x 18. 4x! 1 = 4x 13. a( b! c) = ab! ac = x! 1 x = f + ( d + e ) = ( f + d) + e 15. a!( b!c) = ( a!b)!c "! 1 % $ ' = 0 # 2 & 16.!5 + 5 = ( x! 9) = 7x! Use the Commutative Property of Multiplication to rewrite the given expression. 24. Use the Distributive Property of Multiplication over Addition. 4! 5! 6= 8x - 24= 25. Is x =!5 a solution to the equation 4x 12 =!2 x + 21 ( )?! Show work and explain in words. page 31

34 Area of a Circle with radius = r. A =!r 2 r Surface area of a rectangular prism with length = l, width = w and height = h. SA = 2wl + 2wh + 2lh Volume of a cone with radius = r and height = h. V = 1 3!r2 h Children s Dose of some Medications given the adult dose = A and the child s age = g (years).! g $ C = # &' A " g +12 % Volume of a Sphere with radius = r V = 4 3!r3 Surface Area of a Cylinder with radius = r and height = h. SA = 2!r 2 + 2!rh Use the appropriate formula from the list above. Show work. Include UNITS in your answers. 26. Find the volume of a sphere with radius 27. Find the surface area of a cylinder with 6cm. radius 2in. and height 5in. 28. Find the surface area of a rectangular prism with length 2ft, width 9ft, and height 4ft. 29. Find the dose of medication for a 4 years old child if the adult dose is 12mg. page 32

35 30. Evaluate the following expressions when a = - 2, b = 6, and c = - 1. a! bc b. 5b! 2c + a(2b + c) a. 2b + c Simplify the following as much as possible. If an expression is already simplified, say so. 31) 7x + 5 3x 32) 6x x 33) ( 12x 5) ( 7x 11) 34) 6w 2 +11w + 8w 2 15w page 33

36 35) 11a 2 b 12ab 2 36) ( 2x 2 3x + 7) (!3x 2 + 4x 7) 37) 4( 6x 3 4x 2 +11) 7( 5x 2 + 9) 38)!3(! 4x y) 39. 9( 4x 2 7x +12) 12( 3x 2 5x 9) page 34

37 Answers: ( 3!1)" 4 +1 = ! ! 8 2 = ! 3" 3 =! 2 ( ) e 12. a 13. g 14. i 15. d 16. b 17. f 18. h 19. e 20. c 21. b 22. g 23. 5! 4! (x 3) 25. If x =!5 were a solution to the equation 4x 12 =!2( x + 21), then 4(!5) 12 =!2((!5) + 21). Let s see if it does: = - 32 and - 2(- 5+21) = - 2(16) = - 32 Since both sides equal - 32, x = - 5 is a solution to the equation 4x 12 = - 2(x + 21). 26. V = 4 3! (6)3 29. = 288! cm 3! 4 $ C = " # 4 +12% & '12 = 4!12 = 3 mg SA = 2! (2) 2 + 2! (2)(5) = 8! + 20! = 28! in 2 30a. (!2)! (6)(!1) 2(6) + (!1) =! !1 = SA = 2(9)(2) + 2(9)(4) + 2(2)(4) = ft 2 30b. 5(6)! 2(!1) + (!2) 2(6) + (!1) [ ] = (!2)(12!1) = 32 +!2(11) = 32! 22 = x x x w 2 4w 35. simplified already x 36. 5x 2! 7x x 3! 51x 2! x + 3y page 35