Sharpening your algebra skills: Summer practice for students. Updated 2009

Transcription

1 Sharpening your algebra skills: Summer practice for students Updated 009 1

2 Summer Work: Enhancing Your Algebra Skills in Preparation for a Successful Year at Asheville School The mission of the Asheville School Mathematics Department is to ensure that you leave our school with the best possible education in mathematics. To be successful in each of our courses, it is paramount that you have acquired the skills necessary to enter the course. In order for you to be successful in Algebra and beyond, your teacher has recommended that you do summer work to improve your basic algebra skills. What follows is a collection of 14 topical units; we recommend you work under the guidance of a mathematics tutor. These chapters were developed by Asheville School faculty to help you enhance your algebra skills. Please try the problems (give full effort), check your answers (the answers are provided at the end of the packet), and then review the material with your tutor if you do not get a problem correct. We are confident that your math skills will improve tremendously if you take this work seriously over the summer, and this will in turn lead to a successful year next year. Please let us know if we can help you! Sincerely, The Asheville School Mathematics Department: Pam Reid, Steve Butera, Liz Wolfe, Larry Kollath, Varghese Alexander, Mary Lou Gillum, and Mike Hill (Chair)

3 TABLE OF CONTENTS Section Page I. Fractions, Decimals, and percent 4 II. Order of Operations and Evaluation/Substitution 4 III. Translating Words into Symbols and Translating Sentences into Equations 8 IV. Opposites & Absolute Values V. Basic Algebraic Properties and Operations with Real Numbers 7 VI. Solving Basic Equations 47 VII. Exponents and Polynomials 56 VIII. Factoring 66 IX. Fractions 80 X. Ratio and Proportion 9 XI. Systems of Linear Equations 98 XII. Inequalities 106 XIII. Radicals 111 XIV. Quadratics 118

4 SECTION I Fractions Decimals Percent 4

5 Name Readiness Evaluation: Fractions and Mixed Numbers Express the following in lowest terms. 1) 7 ) ) ) ) ) 80 4 Simplify 1 7) ) ) ) ) ) 6-4 1) ) ) From take

6 16) ) 1 18) ) ) 1) ) Divide 4 1 by 5 ) 5 4 H16 8K 5 F I 4) )

7 Worksheet: Addition of Fractions & Mixed Numbers. 1) ) ) ) 5 + 5) + + 6) ) ) ) ) ) ) )

8 Worksheet: Subtraction of Fractions & Mixed Numbers 1) 7 11 ) 5 ) ) 9 1 5) ) ) 9 8) 7 1 9) ) ) ) ) F H I K - 8

9 Worksheet: Multiplication of Fractions & Mixed Numbers. 1) 5 5 ) ) ) ) 7 6) ) 1 1 8) 7 9) ) 1 11) 1 1 1) ) 14)

10 Worksheet: Division of Fraction & Mixed Numbers 1) 1 4 ) 5 ) 4) ) 15 6) 5 7 7) 8 8) ) 10) 14 11) ) H 8 16K F I F I F ) 4 14) 15) H 4K H 1 4K I F H I K F H 16) I K 17) ) ) ) ) )

11 Name Final Assessment: Fractions and Mixed Numbers Simplify to lowest terms. 1) 18 4 ) ) ) Simplify. 5 5) ) ) 1-1 8) ) F H I ) 11) ) 6 K ) 14) ) 5 16) ) ) )

12 Readiness Evaluation: Decimals Write each of the following as a decimal numeral. 1) Three tenths. ) Seven hundredths. ) Ninety-four thousandths. 4) Two hundred sixty and three hundred forty-seven ten-thousandths. Round to the nearest tenth. 5).6 6) ) 0.6 Round to the nearest hundredth. 8) ) ) $5.475 Simplify. 11) ) ) ) ) ) Subtract 0.08 from 0. 17) ) ) (0.0)(0.) 0) (0.1)(0.04) 1) (0.059)(0.064) ) (70.84)(0.04) ) 0.86 of $ 9.57 correct to the nearest cent. 4) ) ) ) to the nearest thousandth. 1

13 Worksheet: Addition of decimals Simplify. 1) ) ) ) ) ) $7.6 + $ $ $ $.5 7) ) ) )

14 Worksheet: Subtraction of decimals Simplify. 1) ) ) ) ) ) ) $6.80 $17.4 8) ) ) $500 - $,

15 Worksheet: Multiplication of decimals Simplify. 1) (5)(0.4) ) (0.0007)(5.) ) (8.7)(0.48) 4) (70.4)(0.04) 5) (0.000)(0.) 6) (0.0167)(1.8) 7) 0.75 of $4.00 8) 0.45 of 00 9).5 of $56 10) 1.5 of $

16 Worksheet: Division of decimals Simplify. 1) 6.9 ) ) 1.6 4) ) ) Divide by 100 9) 8 10) ) Divide by ) 75 1) 85 14) 0. Change to a decimal. 7 15) 10 16) 1 17) 5 18) ) 8 0)

17 Final Assessment: Decimals Write each of the following as a decimal numeral ) ) ) ) ) ,000 6) Thirty-six thousandths. 7) One hundred and five tenths. Simplify. 8) ) ) ) ) ) from take ) ) ) ) ) $ Round to nearest cent: 19) $ ) ) ) Round to nearest hundredth: ).75 4) ), ,000 17

18 Readiness Evaluation: Percents. Complete the following table of equivalents: Percent Decimal Common Fraction 1) 0% ) 0.5 ) 75% 4) 1 5) 7 1 % 6) 0.4 7) 5 6 8) 100% 9) 150% 10) 0.5% Find the answer to each of the following. 11) What number is % of 49? 1) 0 is what % of 80? 1) 80% of what number is 8? 14) 87 1 % of $4000 is what number? 15) What percent of 75 is.4? 16) 0 is what % of 5? 17) What amount is % of $70? 18) What amount is 1 % of $5000? 19) What % of 1.4 is.5? 0) 1.5% of what number is 1.5? 18

19 Worksheet: Finding a Percent of a Number 1) 48% of 85 ) 60% of 50 ) 6% of 6.8 4) 5 1 % of 8 5).5% of $000 6) 00% of $1000 7) 1 % of 500 8) % of ) 6 1 % of 80 10) 8 1 % of 96 11) 66 % of 807 1) 6 1 % of

20 Worksheet: Finding Equivalent Fractions, Decimals, and Percents Express each of the following fractions as a percent: ) ) ) ) 1 5 5) 1 8 6) 1 7) 1 5 8) ) ) ) 8 1) 0 5 Express each of the following decimals as a percent: 1) ) ) 0.15 Express each percent as a common fraction: 16) 50% 17) 60% 18) 0.5% 19) 16 % 0) 1 % 1) 8 1 % ) 7 1 % ) 6 1 % 4) 87 1 % 0

21 Worksheet: Finding What Percent One Number is of Another Number Find the answer to each of the following: 1) What % of 100 is 9? ) What % of 4 is? ) What % of 7 is 5? 4) 5 is what % of 50? 5) What % of 4 is 4? 6) 50 is what % of 50? 7) 8 is what % of? 8) What % of 75 is 105? 9) What % of 48 is 80? 10) 6 is what % of 9? 11) 1 is what 5 of 600? 1) What % of.8 is 0.7? 1) What % of 00 is 0.5? 1

22 Worksheet: Find The Number When a Percent of It is Known Find the following missing numbers: 1) 18% of what number is 9? ) 7 is 90% of what number? ) 50% of what number is 97? 4) 00 is 500% of what number? 5) 5 1 % of what number is 0? 6) 680 is 4 1 % of what number? 4 7) 175% of what amount is $48?

23 Final Assessment: Percents Write as a fraction in simplest form. 1) 5% ) 7 1 % ) 16% 4) 150% 5) 1 % 6) 7% Express as a decimal. 7) 5% 8) 4% 9) 0.5% 10) 9 1 % Find the answer to each of the following: 11) % of 00 is what number? 1) 6 1 % of 500 is what number? 4 1) 99 is 180% of what number? 14).4 is 45% of what number?

24 SECTION II Order of Operations & Evaluation/Substitution 4

25 Readiness Evaluation: Order of Operations & Evaluation/Substitution a f a f Simplify. 1) 10-4 ) ) a f a f a f a f 4) ) ) Evaluate if: a =, b =, x = 8, and y = 5 a f 7) x + ay 8) 5 - a y 9) a - b 1 10) 4 11) 1 axy 1) x a - b x + a f Evaluate each expression if: t = 6, x =, y = 4, and z = 5 a f a f a f 1) xy+ z 14) 4 xz + y 15) y y + z z 16) 6y - xy 17) 9x + z x+z 18) 10t - z 10 t - z a f a f a f 19) x + 4 y + z 0) 5y + 6z t y 5

26 Simplify each expression. Worksheet: Simplifying & Evaluating Expressions. a f a f a f 1) ) ) ) a17 - f a f a f 5) 6) 5 6- a f Evaluate each expression if: a = 1, b =, and c = 7) 6a 8) 7b 9) c - 10) 9 - b 11) af 5c - 4 1) b + af ac af a f a f a f 1) a + bc 14) a -1 15) a + b c 16) a c - b 6

27 Final Assessment: Order of Operations & Evaluation of Expressions Simplify each expression. 1) ) ) a9 + 5f 4) a f ) a f Evaluate each expression if: t = 6, x +, y = 4, and z = 5 a f a f 6) 18-4 x 7) 9xyz - t 8) 9x zy - t 9) 5ay - 4x f 10) z + 5ay - x f 11) t - 7z ay + xf 7

28 SECTION III Translating Words into Symbols & Translating Sentences into Equations 8

29 Readiness Evaluation: Translating Words into Symbols and Translating Sentences into Equations. Translate each phrase into a variable expression. Use n for the number. 1) Eight times a number. ) The product of three and a number. ) A number decreased by four. 4) A number divided by five. 5) Nine less than half a number. 6) Nine more than twice a number. Complete each statement with a variable expression. 7) A rectangle has a width of 6 units and a length of x units. Its area is square units. 8) The Golden Gate Bridge was built n years ago. Three years from now it will have been standing years. 9) Dale has twice as much money as Leo. If Dale has d dollars, then Leo has dollars. 10) Two numbers differ by 1. If the smaller number is s, then the larger number is. Translate each sentence into an equation. 11) Eight is five less than twice the number x. 1) Forty decreased by the number m is ) The product of 1 and the quantity one less than the number d is ) The product of 7 and the sum of twice the number x and is ) Ten times x is twice the sum of x and eight. 9

30 Worksheet: Translating Words into Symbols. Translate each phrase into a variable expression. Use n for the number. 1) A number decreased by eleven. ) The product of eleven and a number. ) The quotient of 17 and a number. 4) Eleven more than one third of a number. 5) Ten times the sum of a number and nine. 6) Seven decreased by three times a number. 7) Five times the sum of a number and two. 8) Six times the difference of a number and five. Complete each statement with a variable expression. 9) The sum of two numbers is 17. If one number is x, then the other number is. 10) There are twelve fewer boys than girls. a) If the number of girls is g, then there are boys. b) If the number of boys is b, then there are girls. 0

31 Worksheet: Translating Sentences into Equations. Translate each sentence into an equation. 1) The quotient of the number b and four is eight. ) Three fourths of the number x is 19. ) The product of 1 and the quantity one less than the number d is 84. 4) One half of the sum of three and a number n is four. 5) Four less than twice a number x is nine. 6) Ten times x is twice the sum of x and eight. 7) Three diminished by twice a number n is eight. 8) Three times the number which is x less than two is eight. 9) Two times the number which is three less than x is eight. 10) Three times the quantity two less than x is eight. 1

32 Final Assessment: Translating Words into Symbols and Translating Sentences into Equations. Translate each phrase into a variable expression. 1) A rectangle has a width of y and a length of 1. Its perimeter is. ) Al earns (p + ) dollars per hour. In eight hours, he earns dollars. ) A sports arena was d years old 15 years ago. It is now years old. 4) Workers on an assembly line produce (x + 10) cars each day. In five days the can produce cars. Translate each sentence into an equation. 5) The product of 58 and the number n is one. 6) Six less than a number is twelve. 7) Twice a number increased by 18 is five times the number. 8) Three times two decreased by x is eight. 9) Two times the number, which is three less than x, is eight. 10) Twice a number x is two more than x.

33 SECTION IV Opposites & Absolute Values

34 Readiness Evaluation: Opposites & Absolute Value Name the opposite and the absolute value of each number. 1) 7 ) -1 ) 4 4) 0 5) 6.5 6) - 1 Simplify. af a f af 7) - - 8) 1 9) ) af 11) ) -14 1) ) ) ) ) Complete using ine of the symbols <, >, or =, to make a true statement. a f a f 18) ) Translate into symbols. 0) Four is less than the absolute value of negative ten. Solve each equation over the set of real numbers. If there is no solution, write no solution 1) p = ) b = 9 ) -x = 5 Evaluate each expression if: a = 15., b = -, and c = ) a + -b 5) a b a f a f+ c 4

35 Worksheet: Opposites and absolute values. Name the opposite and the absolute value of each number. 1) 5 ) - 9 ) ) 0 Simplify. 1 5) - 6) -F- 1 I H K 7) -a9 + 8 f 8) -a0 f 9) 14 10) ) 8 -af -8 1) Complete each statement. 1) If n is a negative number, then n is a number. 14) A real number that is its own opposite is. Evaluate if : a = 15., b = -, and c = ) b a + c 5

36 Simplify. Final Assessment: Opposites and Absolute Value 1) -.4 ) -0 ) -a10-8f 4) ) -af ) ) ) ) - + Solve each equation over the set of real numbers. If there is no solution, write no solution 10) n = 0 11) -z = 0. 1) x = 4 1) -x = 1 Evaluate if : a = 15., b = -, and c = -1.7 a f a f 14) a c + b 15) 4a - b c 6

37 SECTION V Basic Algebraic Properties & Operations with Real Numbers 7

38 For all real numbers a, b, and c: Basic Algebraic Properties 1) Closure Properties: a + b is a unique real number ab is a unique real number ) Commutative Properties: a + b = b+ a ab = ba ) Associative Properties: (a + b) + c = a + (b + c) (ab)c = a(bc) 4) Properties of Equality: Reflexive Property a = a Symmetric Property If a = b, then b = a Transitive Property If a = b, and b = c, then a = c 5) Identity Properties: Addition- There is a unique real number 0 such that for every real number a, a + 0 = a and 0 + a = a. Multiplication- There is a unique real number 1 such that for every real number a, a (1) = a and (1) a = a. 6) Property of Opposites: For every real number a, there is a unique real number a such that a + (-a) = 0 and (-a)+ a = 0. 7) Property of the Opposite of a Sum: -(a + b) = -a + (-b) 8) The Distributive Properties: a(b + c) = ab + ac and (b + c)a = ba + bc a(b - c) = ab - ac and (b - c)a = ba - bc Applying the symmetric property of equality, the following are also true: ab + bc = a(b + c) and ba + bc = (b + c)a ab - bc = a(b - c) and ba - bc = (b - c)a 1 1 9) Multiplication Property of Zero: a = 1 and a = 1a 0=0 and 0 a=0 a a 8

39 For all real numbers a, b, and c: 10) Multiplication Property of -1: a(-1) = -a and (-1)a = -a 11) Property of Opposites in Products: (-a)b = -ab, a(-b) = -ab and (-a)(-b) = ab. 1) Property of Reciprocals: For every nonzero real number a, there is a unique real number such that a = 1 and a = 1 a a a 1) Property of the Reciprocal of the Opposite of a Number: For every nonzero real 1 1 number a: a = a 14) Property of the Reciprocal of a Product: For all nonzero number a and b, = ab a b 9

40 Readiness Evaluation: Real Numbers Simplify each expression. 1) ) ) a b + a f a f a f a f 4) ) ) ) a-7 f+ a-6 f+ 75 8) Evaluate - x + y + a- f, if x = - & y = -4 9) 51-a7-104 f 10) -.6 -a-9.15 f 11) -ar - 4f a r f a f a f a f a f a f af j 1) ) a a ) j + + a fa fa f a f a f a fa f a f 15) ) - 7 -a + b - -b 17) ) F -7 - t 1 19) x + 66y ) Ia ff I F Ia f H K H K H K F H I K 1 1 1) 8 F 75-1 ) -j a I 4 8b 1 H F k K H ai K a f F I H - K ) 66z -14z if z 0 4) If x 0, what is the reciprocal of - x 5) Evaluate: b -7c a-c if a = -, b = -1 & c = 40

41 Worksheet: Addition of Real Numbers. Simplify. 1) ) ) a f a f a f a f a f a f+ a f+ F I F I H K H K + F H 4) ) ) I K 7) F H 8) I K F H 9) I K F H I K af afa f 10) ) a a 1) x x + 1) - b + -a + aa + b f 14) -a-x + y f + y Evaluate each expression if: x =, y = 6, and z = -4 15) - z + a-8 f + y 16) x + a-y f + z 17) - y + a-11 f+ x 18) - x + z + a-8f 41

42 Worksheet: Subtraction of Real Numbers. a f Simplify. 1) 5-1 ) ) a f 4) ) 174 -a-4 f 6) 1.91-a-1.0f 7) 4 less than -1 8) 1 -a7-61 f 9) 4 -a56-87f a f a f a f a f a f a f 10) ) ) a f a f 1) ) ) k k 1 16) The difference between 81 and -6, decreased by ) The sum of 8 and y subtracted from z. Evaluate each expression if: a = 4, b = 5, and c = -1 18) c - a - b 19) a - c - a b 0) a - c a b b g b g 4

43 Worksheet: Multiplication of Real Numbers and The Distributive Property. a f a f a f Simplify. 1) 4 x + + ) a ) 9 n - + 4n a f a f a f a f a f 4) 5 7y - z + 4 5) 7 c + d c 6) 4 5x + y y 1 7) a-7afa-5cfa4 f 8) a-xfa-yfa-z f 9) -aa + b - f a fa f a fa fa fa f a fa fa fa f 10) y 11) ) -4 5 a f a fa f a f a f 1) x + 5y + 7x - y 14) - q + w 7 w - q a f a f a f a f 15) - 4 e + f e+ 5 f 16) -7 r + s r 7 s + r 4

44 17) a f F H ) I K F H I K F H I K F H 19) I K 0) F H I F KH 1 I K F H I F K a f I H K a f F H 1) 1xy 1 1 ) 5x if x 0 ) 8ab - 1 4) 6ac x I K 1 5) ) a a f a m kf 7) a 7 8u +10v f - 1 a15u - v f + F H I K F I F I a f a f ) 6 x y y x 9) m 16 84m + 8 H K H 7 K 8 4 a f F H 1 0) r+ 4s + 7 s r I K 44

45 Simplify. Worksheet: Division of Real Numbers. a f F I H K 4) 0 1) ) -1 - ) ) ) ) ) a ff H I K 9) -4 -y 10) 156a 1 11) -54x 6 1) -144y -4 Evaluate each expression if: a =,b=-1,c=,and d=6 1) b+5c -d 14) abc - d abd 15) a fa af c+a c b +d 45

46 Final Assessment: Real Numbers Simplify. 1) 16 + p + q + 5 ) ) af a f F H I K 4) n ) ) a f a f a f a f a fa f a fa f 7) ) x x ) a f a f a f 10) 6x ) 5 b ) 7 c - 4d + 6 a f a f a fa fa fa f 1) ) ) ) 7x - y + x + y 17) F H I F Ka f H I K 18) ) a f 4w -56m + 49n 0) -7 1) What is the reciprocal of -1 46

47 SECTION VI Solving Basic Equations 47

48 Readiness Evaluation: Solving Basic Equations Solve. 1) x 7 =1 ) t - 5 = -18 ) x + 6 = 4 4) 8 = -x +18 5) x =1 6) a =15 7) x + 4 =1 8) x + 6 = 9) - 4x = 44 10) x =1 11) x = 6 1) 1 y = 1 1) -x 5 = 0 14) - 5 x =10 15) 8 x = 16) x 8 = 4 48

49 17) -y = 18) 5x + 7 = -8 19) x 7 =17 0) x+5 = 7 1) x +5-7x =15 ) - 1 (x + 4) =16 ) - = 4(x + 7) -15 4) (x -1) - (x - 5) +x = 0 5) 4x - 8 = x 6) 4(x +) =14 - ( - x) 7) x +[1- (x +)] = x 8) 1 (0 4x) = 6 x 4 49

50 Worksheet: Transforming Equations: Addition & Subtraction Solve. 1) y - 9 =17 ) n = 6 ) t - 5 = -0 4) x - 6 =18 5) x - 6 =14-8 6) x - 57 = -67 7) x = -.8 8) x = 1 4 9) -1 + x = 1 10) x + = 1 11) x - 4 =11 1) y = 8 1) x + 6 =1 14) ( x 8)+15 = 7 50

51 Worksheet: Transforming Equations: Multiplication & Division Solve. 1) 5y = 65 ) x = -7 ) 1 x =1 4) x 5 = 7 5) x =17 6) 5 y =10 7) 4 x = 60 8) - x = 9) - 7 x = 4 10) 5 x = ) y = ) - n 4 = 1) - 7 x = 4 14) x = 6 15) 5x =10 16) 4 7 x =16 51

52 Worksheet: Using Several Transformations Solve. 1) 5x + 7 = -8 ) - x +5 =19 ) - 8y -11 =1 4) 1 x + 7 = 6 5) 4y 5 = 8 6) y +5-4y = -10 7) = n - n +5n 8) (x + ) =1 5 9) (x +8) - 9 = 5 10) = 7(x - ) ) x -1 = 5 1) 4y + 7 = 9 1) (p +10) =18 14) 9-4 (x ) =1 5 15) (5 - y) +(6 - y) - (5 - y) = 0 16) 8x = x ) x = 7-15x 18) x 5

53 19) - 7x = -1x -15x 0) 6-4y = y 1) -11x 7 = 5x ) 1 (1 6y) = 4 y ) (x - ) - 4 = (x - ) 4) 6x - ( - x) = 4(x -1) 5) 5x +(1- x) = (x -1) 6) + 4(y +) = y + (y + 4) 5

54 Final Assessment: Solving Basic Equations Solve. 1) y +5 =10 ) 7= x -1 ) x +51 = 8 4) 1 1 x = 65 5) -19v = ) - x 1 = 5 7) 1y - 7 =11 8) x + 6 =16 9) x = 0 10) - 7 (w 16) = ) 7(x - 6) = - +6x 1) 6(m-1) =6(m +) 1) 1 (x + 4) = x 14) 16 = 4 x +1 15) 9( - y) = y 16) (x -4)=6(x-) 17) x = 18) y +6 = 54

55 19) - ( y +) = -6 0) 0 = 8-w 5 1) 10-4 x = - ) - 1 x=51 ) 1 x+ =1 4) (x - ) +6 = - 55

56 SECTION VII Exponents and Polynomials 56

57 Readiness Evaluation: Exponents and Polynomials Simplify: 1) 5 ) 5 ) (-) 4 4) ( +) 5) - 6) (-) 4 Evaluate if a= and x=: 7) x 4 a 8) ( a + x) Simplify: 9) ) 6 [ 5 ( ) ] 11) ( 1 10 )+ ( 7 10 )+ ( 7 10)+ 6 1) ( x - 5y +)+ ( 5x + 6y 7) 1) ( 5x - t - 7) ( x t ) Solve: 14) ( 4u - 6)= 4u ( ) ( u 8) 57

58 Simplify: 15) n n 5 16) x x 4 x 17) ( x )( 5x 5 ) 18) ( y z)y ( z ) 19) ( 4x 5 )( x ) 0) (-xy ) x ( y) 1) (-x y )( xy )( x y) ) t 4 t ) 15a b 8ab 10 4) ( 8c ) d ( ) 1 4 cd 5) x () 4 6) ( t ) 4 7) ( c) ( c) 58

59 8) 1 p q [( ) ] ( pq ) 4 9) x 0) x( x x + 4) 1) 1 s t( 4t 10st + 6s ) Multiply: ) ( y + ) ( y + ) ) ( z - ) ( z + ) 4) ( x +) ( x + x + 5) 5) ( n - 5) ( n n ) Solve: 6) ( x +) ( x 5)= ( x 1) ( x ) 7) ( x +5) ( x )= ( x 1)6x ( + 5) 59

60 Simplify: Worksheet: Exponents, Multiplying Monomials, Powers of Monomials 1) 5 ) ( 5 ) ) 7 + 4) ( 7 +) 5) -5 6) ( -5) 7) ( 1 10 )+ ( 4 10 ) ( )+ 8) [ 5 + ( ) ] 7 [ ( )] 9) ( 4 4 ) ) a a 11) ( y z) ( y z ) 1) ( 4x 5 ) x ( ) 1) ( ab )( 5a b )( a ) 60

61 14) 4h k 7 1hk 5 15) x ( ) 1 6 x 8x ( ) 16) ( 5b 4 ) a ( b) a ( ) 17) x 4 x 7 18) ( x 4 ) 7 19) (-5a 4 ) 0) -( 5a 4 ) 1) ( 4x ) ) (-y ) ) ( r s 4 ) 4 4) ( x y ) 4 ( xy ) 1 5) 10 x y ( 10y) 4 61

62 Worksheet: Adding, Subtracting, and Multiplying Polynomials Simplify: 1) x - y - x - y ) ( x - 5y +)+ ( 5x + 6y 7) ) ( x - 5) ( x ) 4) ( y y 5) ( y 7y + 4) 5) ( a ab ) ( a 4ab b ) Solve: 6) ( 4y - ) ( 4 y)= ( y + ) 7) ( 4u - 6)= 4u ( ) ( u 8) Simplify: 8) x( x x + 4) 9) 1 x y( 4y 10xy + 6x ) 6

63 10) 6r ( r 1) r 4r 5r ( ) 11) 8n - [ n - ( - n) ] Multiply: 1) ( n + 7) ( n + 5) 1) ( r - ) ( r + 6) 14) ( x + 7) ( x 7) 15) ( x + ) ( x + x + 4) 16) ( y - 5) ( y y 7) 17) ( 5 + x) ( x 5x + 4) 6

64 Final Assessment: Exponents and Polynomials Simplify: 1) 9-4 ) ( xy + 4x y 6)+ ( 5xy 5x y 7) ) x 6 1 x 6 4) ( a 4 b)5a ( b)a ( ) ( ) 1 6) 9n 5) -x y 4 n 4 1 [ ] 8) 4n 7) -616a-8( a-) n 4 Solve: 9) x - ( 15x - 6)=104 10) 6- ( n - )=1 64

65 Multiply: 11) ( 4x - ) ( x 4) 1) ( c - 6) ( c + c + ) 1) ( a - b) ( a + ab + b ) 14) ( x +5) ( x + 5) ( x + 5) Simplify: 15) ( 5n 4 )n n ( n ) 16) ( 5y ) ( x y) 17) ( x) ( xy) ( xy) ( x) 65

66 SECTION VIII Factoring 66

67 Readiness Evaluation: Factoring Give the prime factorization of each number: 1) 4 ) 00 ) 40 Simplify each fraction: 5-70m 4) 4mn 7 (- x) 4 5) 9x - 65r 6) r 1r 4 5r Write each product as a polynomial: 4 4 ( 5m -1)( 6m 5) 8) ( c + c )( c ) 7) c ( x - 9) 10) ( 4m - 6 ) 9) n Factor completely. If the polynomial is not factorable, write prime: 11) b b + 1) x x + 4 1) a 6a 40 67

68 68 16) + 6r 7-15) 15 6 ) 14 y y x r y y + m m m x ) ) 4 + z 0) ) z x x Solve: m m ) x x ) = = ) = x

69 Give all the positive factors of each number: Worksheet: Factoring Integers 1) 1 ) 0 ) 68 4) 7 State whether or not the number is prime. Give the prime factorization of the number: 5) 1 6) 7) 51 8) 81 9) 9 10) 6 11) 100 1) 47 Find the GCF of each pair of numbers: 1) 1 and 18 14) and 46 15) 16 and 4 16) 1 and 4 17) 18 and

70 Find the GCF of the given monomials: Worksheet: Dividing Monomials 4 1) x, 9x ) 15a, 1a ) p q,p q 4) 7xy, 14x y 5 ) 6s t, 9st 6 ) 0ax, 0abx Simplify. Assume that no denominator equals 0: 8 6 7) ) 9) w 10) 4w a 11) 6a 1 ) 4b 1b 1) -6 p 1 p 5 5x y 14) xy 15 ) 5 mn m n 16 ) 5 ( t) ( t) 70

71 17) s 18 ) ( s) ( ) -x 4 x y 71

72 7 Worksheet: Monomials Factors of Polynomials Divide: y y xy+ ) t- ) a ) 1 pq q p pq ) r r r r ) x x ) q p q p q p q p ) s r s r s r s r ) + + Factor: c c ) b+ a ) ) 11 y x y x ) - b a ab ) 1 c b a d c ab ) z xy w z y wx +

73 Write each product as a binomial: 1 ) Worksheet: Factoring the Differences of Two Squares ( y-7)( y + 7) ) ( 4+x)( 4 x) ) ( x+ y)( x y) 4 ) ( x 9y)( x + 9y) 5) ( ab-c )( ab + c ) 6 ) ( xy+ z)( xy z) Factor: 7) b 6 8 ) 9a ) 16w 6z 10) 5z 1 11) 81n 11 1 ) 1-9a -y 1) ) 49a 9b 15 ) 64u 5v -c )16 17 ) 65x 1 18 ) u 81v 8 19) x y 8 0 ) m

74 Worksheet: Squares of Binomials and Perfect Square Trinomials Write each square as a trinomial: ( n+ 5) ) ( a-9) ) ( 4 1 ) 1) u- ( 5n- 4) 5 ) ( 5p- 6 ) 4) q 6 ) ( mn + ) 7) ( ab + c ) (- 4m n) 9 ) ( 9 p 10) 8 ) + Factor: 10) y + 6y ) x 4x + 4 a ) 16a 64 1) 4s 6st 81t 14) 8x + 8x + 15 ) a 18a + 7 y ) 14y 49y 17 ) 8u 4u v 18uv 74

75 Worksheet: Factoring x +bx+c and ax +bx+c Where c is Positive or Negative Factor: 1) x + 5x + 4 ) q +16q +15 ) x + 0x + 6 4) z +16z + 9 5) 75 +0r +r 6) a +10ab + 4b 7) y + 5y 6 8) x 6x 16 9) y 1y 7 10) 1+15c - 4c 11) x + 7x + 1) p + 7p 6 1) 4c + 4c 14) 9m 5mn 6n 15) 18z +19z 1 16) r -18r 17) 1c + c 4 18) x +10xy 8y 75

76 Worksheet: Using Several Methods of Factoring Factor Completely: 1) 5a +10ab + 5b ) 6c +18cd +1d ) 4m m 4) xy 7x 5) y 4 y y 6) - n 4 n n 7) - 41a a 8) 80-10p + 45p 9) 8p q 18pq 10) 6u v 11u v 10u v 11) u 5 7u 4u 1) 16c

77 Worksheet: Solving Equations by Factoring Solve: 1) ( y + 5) ( y 7)= 0 ) 15n( n +15)= 0 ) ( t - ) ( t )= 0 4) x( x +1) ( x + 5)= 0 5) y y + = 0 6) m 6 =16m 7) y =16y 8) 4x 9 = 0 9) 5x 90x = 81 10) 4x 1x + 8x = 0 11) y 4 10y + 9 = 0 1) 9x + 9x = 0x 77

78 Final Assessment: Factoring Factor to primes: 1) 66 ) 50 ) 504 Simplify. Assume that no denominator equals zero. 4) (-xy ) 6x y 5) 4a4 b 5abc ab c 6) -( ab +c) -9( ab +c) 7) 16g5 h 8g 6 h + 4g 7 4g 5 8) 1t t 9t t Express each product as a polynomial: 9) ( m - ) ( m 1) 10) c( 4c +5) ( c ) 11) ( 5e - f) ( 7e + 4 f ) 1) ( z + 7) ( z 7) 1) ( 5a - 8b)5a ( + 8b) 14) ( r - ) 78

79 15) ( s +6t) 16) (-j + k) 17) ( mn p) Factor completely: 18) 89x 6y 19) 4v 6 9 0) 5u 10u +1 1) n 17n + 60 ) r +18r 6 ) 1-11ab -1a b 4) 7x 11xy 6y 5) - 6c + 8c + 8c 6) m 4 4m 5 Solve: 7) 7x = x 8) 18y + 8 = 4y 9) z +10z = 4z 79

80 SECTION IX Fractions 80

81 Readiness Evaluation: Fractions Simplify. Give the restrictions on the variables for a + 4ab 1.) 6a + 1ab + 6b.) x 98 8x + 4x.) 9c4 d 7de 14ce 5) 4.)(a 6 7c a 6a6 5 a 5.) a b 6a b 6.) 4r + 4s 8rs 8r + 8s 4r s 7.) x + x x + 5x x x x 8.) ab a b + a b a 81

82 9.) 1 6n 5 4n 10.) x 45 x ) z + z + z z + 4 z 4 1.) c + 1 c 1 + Divide. Write your answer as a polynomial or mixed expression. 1.) + 4x + 6x x 1 14.) 6a 11a + 1 a 1 8

83 Worksheet 1 Simplifying Fractions Simplify. State the restrictions on the variable. 1.) m + 9 m +.) n + 8 n + 1.) a 16 a 4 4.) b 9 b + 5.) x x + 1 x 1 x ) 6 x 14 c 7.) 7 c c d 8.) c + d 9.) t 1 1 t (y 8) + 5) 10.) 11.)(x (y 8) 5 + x 1.) (4 x)(x 9) (x 4)(x ) 8

84 Multiply and simplify Worksheet Multiplying Fractions 1.) ) ) ) a b b a 5.) x x 14 6.) n 6 16 n 7.) 5y 6 y 8.) a a 4 9.)(c) 4 c b 10.) (a) a b (x 1) 4 11.) 8 x 1 1.) n n n 4 n Simplify: 5a 1.) b x 14.) y 15.) c 16.) 4 n 84

85 Worksheet Dividing Fractions Simplify. Give your answer in simplest form. 1.) ) 4y 7 y 14.) 5n 7 10n 1 4.) x 4y x 1xy 5.) a b 4 ab 6.) 4y 5 7.) + y 6 1+ y 9 8.) y 16 4y y ) a b 1 b a 10.) y 1 y 4 y + y y y 11.) x 11x x x + 4x 6x 96x 85

86 Simplify Worksheet 4 Adding and Subtracting Fractions 1.) ) ) x 8 + x 8 4.) 7a 1 a 1 5.) 4 x 1 x 6.) 7 x + x 7.) x + + x x + n 8.) n n + 4 x 9.) x x + 7 x 10.) x 1 1 x ) a b + 1 b a 1.) x x 1.) x + x 8 86

87 14.) y 4 y 6 15.) 6 a + a 16.) 5 n 4 n 17.) 6y 5 y ) n 4 n ) c + c ) x x 6 y 1.) y 5 y y 5.) x 11 x 9 x 7 x x 87

88 Worksheet 5 Mixed Expressions State each expression as a fraction in simplest form. 1.) 1 8.)5.) ) )1+ 1 x 6.) + 4 a 7.)n n 8.) y + y 9.) a b 10.) 1 x ) x 1 1.) b b ) + x )4 1 y + n 15.) n )x 8 x + 1 6x x

89 Worksheet 6 Polynomial Long Division Divide. Write the answer as a polynomial or mixed expression. 1.) x + 5x + 6 x +.) y y + 5 y + 1.) y 9 y ) a + a + a + 4 a 1 5.) n 1 n 1 6.) 10n n + 8n 4n

90 Final Assessment Fractions Simplify. Give the restrictions on the variable. 5x ) x 49.) x 6x 4 x + x 8 64 n.) n 4n Express in simplest form. 4.) n 7 5.) a b 7ab 54 6a ) a 6 y 7.) 6a a x y x 8.) x 1 + x 6 9.) 6n + n 5 4n n 90

91 Write each expression as a fraction in simplest form. 10.)1 n 5 11.)4x x + 1 x 1 x 1.) x + + x + 1 Divide. Write the answer as a polynomial or a mixed expression. 45 1n + n 1.) n 5 14.) x x 5x x

92 SECTION X Ratio and Proportion 9

93 Readiness Evaluation: Ratio and Proportion State the ratio in simplest form. 1.) 5:15.) 4x:6x.) 5m to 5cm 4.) Π(r) Πr 5.) 1kg to 50g Solve. 6.) x = 4 7.) a + 5 = 7 10a 1 8.) x + 5 8x = 9.) b + 4 6b = b b + 8 Simplify. Give the answers using positive exponents. 10.) e e 11.) g g 1 1.)x x 9

94 Write a ratio in simplest form. Worksheet Ratio 1.) 14 : 1.) 6y : 9y.) (s) s.) 7m5 45m 5.)64a b 16ab 4 7rs5 6.) 1r s 7.) 0min : h 8.) 4h : 45min 9.) 6m : 10cm 10.) 18cm :1.8m 11.) 6wk : days 1.) 5km : 450cm 1.) 1oz : lbs 14.) kg : 90 g 94

95 Solve. 1.) x 5 = 4 Worksheet Proportion.) 7 6 = a.) 18x 1 = ) 1t 7 = ) 15 5 = 4x 15 6.) 4 = 8b 5 7.) x 5 4 = 8.) 6x 7 = 5x x ) = x )x = 5(x ) ) x = 7 x + 95

96 Solve. Worksheet Fractional Equations 1.) w + w 4 = 7 4.) 7m 8 m 5 = 11.) x 5 x = 0 4.) x + 15 x = 5 5.) n + n 4 = n 5 6.) x + 4 x 7 = x ) 1 4 (n + ) 1 6 (n ) = 8.) 1 (x + 4) (x 1) = 96

97 Worksheet Negative Exponents Simplify. Give answers as positive exponents. 1.) 10-1.) 1-1.) 6-4.) 5-5.) 1-6.) 4-7.) ) ( -1 ) 9.) ( -1 ) 10.) ) ) (5x) - 1.) ) c- d 15.) (5x ) 1 16.) (y - ) 97

98 Final Assessment Ratio and Proportion Express each ratio in simplest form. 1.) 1 days : 5 wk.) 6 cm : 0.9 cm Solve..) 104 x = 1 4.) 16 y = ) x 10 = 5x 6 6.) 4a = a 4 7.) y + 5 y + 5 = 10 8.) b 5b = 0 9.)c + c + 6 = 1 c Simplify. Give answers as positive exponents. 10.) ) ) (4m ) 1.) (a - ) 14.) n n 6 98

99 SECTION XI Systems of Linear Equations 99

100 100 Systems of Linear Equations Readiness Evaluation 1. Solve by the graphing method: 9 = + = y x x y. Solve by the substitution method: 11 7 = = + y x y x. Solve by the addition-or-subtraction method: 4 0 = + = + y x y x For Problems 4 and 5, solve by using multiplication with the addition-or-subtraction method: = + = + b a b a = = + b a b a

101 101 Systems of Linear Equations Worksheet: Solving by Graphing Solve each system by the graphing method. 1. x y x y = = = + = x y x y. 8 1 = + = x y x y = + = y x y x = + = y x x y 6. 5 = = x y x y = + = + y x y x

102 Systems of Linear Equations Worksheet: Solving by Substitution Solve each system by the substitution method. 1. y = 6x x + y = 8. x + y = 11 x = 4y. x + y = x = 5 y 4. a b = 1 a = b 5 10

103 Systems of Linear Equations Worksheet: Solving by Addition or Subtraction Solve each system by using addition or subtraction. 1. x + y = 7 5x y = 1. 4y + x = 9 4y x = 7. 1n + m = 18 5n + m = p 18q = 14 15p 18q = x 16y = 1 x + 16y = 19 10

104 Systems of Linear Equations Worksheet: Solving by Using Multiplication with Addition or Subtraction Solve each system by using multiplication with addition or subtraction. 1. c 8d = 7 c + d = 7. x + y = 7 x y = 11. n + 5a = 14 6n + 7a = x + 15t = 10 x + 10t = z 5t + 10 = 0 4z 7t + 5 = 0 104

105 Systems of Linear Equations Final Assessment Solve by the graphing method. 1. x y = 9 x y = 7. y = x 1 x 4y = 10 Solve by using the substitution method.. 8m + n = 5m + n = 7 Solve by using the addition-or-subtraction method. 4. 4x 5y = 0 8x + 5y = p + 4q = 10 p 8q = 6 Solve by using multiplication with the addition-or-subtraction method. 6. x + 5y = 16 5x y = 105

106 SECTION XII Inequalities 106

107 Inequalities Preliminary Evaluation Solve each open sentence and graph its solution set j > k k Solve each open sentence.. 8 c x + 8 > or 7x g n > 1 Graph the solution set or each system. 7. y x y < x y < x y > x 107

108 Inequalities Worksheet: Solving Inequalities Solve each inequality. 1. n 4 > < x 9 d c > ( x 1) > x y ( y ) 108

109 Inequalities Worksheet: Combining Open Sentences Solve each open sentence. Graph the solution set, if there is one. 1. < y r < 4. 8 < n y < 9 or 1 + y > x r > 18 or 1 + r 0 109

110 Inequalities Worksheet: Absolute Value in Open Sentences Solve each open sentence. 1. m 6 = 8. r <. 5. a 4 4. s 7 110

111 Inequalities Worksheet: Systems of Linear Inequalities Graph each pair of inequalities and indicate the solution set of the system with crosshatching or shading. 1. y 5 x 1. y > x x < 1. x > y x y < 4x + 4 y > 4x

112 Inequalities Final Assessment Solve each open sentence and graph its solution set c + 1 < 5. k + 5 or k + 5. y = 4. m 1 7 Graph each inequality in a coordinate plane. 5. x 6. y x Graph the solution set of the system. 7. y < x + y > x 11

113 SECTION XIII Radicals 11

114 Radical Expressions Preliminary Evaluation Simplify, indicating the positive square root of the number in simplest radical form ( 8) Solve for x. Assume x represents a positive number. 1. x + 5 = ( x ) + (7 ) = (x ) 114

115 Simplify. Assume that all variables represent positive real numbers

116 Radical Expressions Worksheet: Radicals Find the positive square root of the number in simplest radical form x b ( 5 )

117 Radical Expressions Final Assessment Find the positive square root of the number in simplest radical form (6 ) ( 9 )

118 SECTION XIV Quadratics 118

119 Using the Quadratic Formula Preliminary Evaluation Use the quadratic formula to solve each equation. Give irrational roots in simplest radical form. 1. s s = 0. y 6y 8 = 0. m + 8m + 7 = 0 4. n 6n 1 = 0 5. z + 8z + 5 = 0 119

120 Using the Quadratic Formula Worksheet Use the quadratic formula to solve each equation. Give irrational roots in simplest radical form. 1. r r 1 = 0. t + t + 1 = 0. 1 s = s 4. 5m + 8m = x = 6x

121 Using the Quadratic Formula Final Assessment Use the quadratic formula to solve each equation. Give irrational roots in simplest radical form. 1. x x = 0. t t 6 = 0. x 8x = x = x