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UNIT 2 i+ ,/ r. r'-]\ ?.: i ."-1 1: n Xi1-,rJ:i. l:.. :1',:":.:"d1 -_- s.:i :: :!. ";i""r=-::r,li{...1 I ?1 ;:-*r;r ri- ::.,,,.'-j i..:; -;,,. 56 uNlr 2 Variable Expressions List of Objectives To evaluate a variable expression To simplify a variable expression using the Properties of Addition To simplify a variableexpression using the Properties of Multiplication To simplify a variable expression using the Distributive Property To simplify general variable expressions To translate a verbal expression into a variable expression given the variable To translate a verbal expression into a variable expression by assigning the variable To translate a verbal expression into a variable expression and then simplify the resulting expression UNIT SECTiON 1 1.1 Objective 2 57 Variable Expressions Evaluating Variable Expressrons To evaluate a variable expression Often we discuss a quantity without knowing its exact value, for example, the price of gold next month, the cost of a new automobile next year, or the tuition cost for next semester. ln algebra, a letter of the alphabet is used to stand for a quantity which is unknown, or which can change or vary. The letter is called a variable. An expression which contains one or more variables is called a variable expression. A variable expression is shown at the right. The expression can be rewritten by writing subtraction as the addition 5y + zxy - x - 3x2 - 3x2 + 1-5y) + 2xy + (-x) + (-7) 7 of the opposite. Note that the expression has 5 addends. The terms of a variable expression are the addends of the expression. The 3x2 expression has 5 terms. The terms 3x2, variable terms. -5y, 2xy, and -x 5 terms 5y + Zxy x - constant variable terms are 7 term The term -7 is a constant lerm, or simply a constant. Each variable term is composed of a numerical coefficient and a variable part (the variable or variables and their exponents). When the numerical coefficient is or -1, the 1 is usually not (x = 1x and -x - -1x). numerical coefficient 3x2 5y + 2xy 1111 [email protected] lx 7 1 wrrtten Variable expressions occur naturally in science. ln a physics lab, a student may discover that a weight of one pound will stretch a spring ] incfr. Two pounds will stretch the spring 1 inch. By experimenting, the student can discover that the distance the spring will stretch is found by multiplying the weight OV l. eV letting W represent the weight attached to the spring, the distance the spring stretches can be represented by the variable expression With a weight of 71l ] W. pounds, the spring will stretch With a weight of 10 pounds, the spring will stretch With a weight of 3 pounds, the spring will stretch tr.* |.tO =]frfz incfres. = 5 inches. t.t = 1] inches. Replacing the variable or variables in a variable expression and then simplifying the resulting numerical expression is called evaluating the variable expression. 58 UN lT 2 Variable Expressions Evaluate ab Example 1 b2 when a - 2 andb - -3. Step 1 Replace each variable in the expression with the number it stands for. ab-b2 2(-3) - (-3)2 Step 2 Use the Order of Operations Agreement to simplify the resulting numerical expression. -6-9 Name the variable terms of the expression 2a2 Solution - - + 5a Example 2 -15 Name the constant term of the expression 6n2 7. + 3n - 4. Your solution 2a2 -5a Example 3 Sofution Evaluate x2 - Example 3xy when X=3andY=-4. 4 Evaluate 2xy + y2 when X=-4andy-2. Your solution - 3xy 32 - 3(3)(-4) e - 3(3X-4) xz e(-4) - (-36) 45 e e Example 5 {;f Evaluate Example *r,en 6 {jf wnen a=5 andb= -3. d=3andb=-4. sorution Evaluate Your solution ++ _ (_4)2 32 - (-4) I - 16 5=:Z-) 3 -t 7 --' Example 7 Evaluate x2 when z-a z-o. Solution I x - - y:31x - 2, y) -1, - 3(x - y) - z2 22-3[2-(-1)]-Sz x2 22 -3(3) -32 4-3(3)-9 4-9-9 -5-9 -14 - z2 Example 8 and Evaluate xs - 2(x * y) + z2 whenx=2,y- -4,and Z=-3. Your solution UNIT 2 59 Variable Expressions 1.1 Exercises Name the terms of the variable expression. Then underline the constant term. 1. 2x2+5x-8 2. -3n2-4n+7 3. 6-a4 Name the variable terms of the expression. Then underline the variable part of each term. 4, 9b2 4ab + - a2 5, 7x'y + 6xy, + 6.5-8n-3n2 1O Name the coefficients of the variable terms 7. -9x +2 x2 8. 12a2-Bab-b2 Evaluate the variable expression when o 9. n3-4n2-n+9 a - 2, b = 3, and c : -4. o I @ o 2b 11. a - 2c 12. 14. -3a + 4b 15. 3b - 16. b2-3 17. -3c + 4 18. .16 -r (2c) 19. 6b -r (-a) 20. bc +- (2a) 21. -2ab -+ c 22. a2 - b2 23. b2 - c? 24. (a + b)2 25. a2 a 6z 26. 2a - (c a 27. (b a)2 29. br-+ 2s. ff - sco 10. 3a + 13. 2c2 -a2 o a E uo E c 6 c o O a)2 - 30. (b - 3c + 2a)z 4c + bc 60 2 UNIT Variable Expressions Evaluate the variable expression when a 31. b+c g2. d-b g4. b +za 35. b-d d - -2, b = 4, c = -'1, and d = 3. 33. 2d+o -a c 36. 2c-d _ad c-a b 37. (b + d)2 - 4a 38. 40. (b 5 41.'b2-2b+4 43. 9!*c - c)2 -r oo. a (d - a)2 - 39. (d-a)2+5 3c 42. a2-5a-6 (-c) ++ 45. 2(b + c) - 2a o 46. 3(b - 4e. taz a) - bc - 2oz abc 52. b-d b-2a 47' bcl-d 50. laa 59. c2 - ab a - 2.7, b 62. (a+b)2-c d E o I ! c o g o c o c) -4bc 51. 2a-b cz +ac + (2a)2 o adTS; 54. d3-3d-9 -fta-ftoa-ac) Evaluate the variable expression when et. lffl e9 - b2-a 53. a3-3a2+a ss. -to * t{"" + oa1 58. (b + c)2 + (a + d)2 I o o 57. (b - 60. 3dc 6)z - - (d - (4c)z = -'1,6,andc=-0.8, 63. E-q" c 6)z UNIT SECTION 2 2.1 Objective 2 61 Variable Expressions Simplifying Variable Expressions To simplify a variable expression using the Properties of Addition ---f,[email protected] + 4 Like terms of a variable expression are the terms with the same variable part. (Since x2 = x. x, x2 and x are not like terms.) -'7x'+ I - x2 Constant terms are like terms, 4 and 9 are like terms. simplily a variable expression, add, or combine, like terms by adding the numerical coefficients. The variable part of the terms remains unchanged. To Simplify 2x + 3x. 2x+3x Add the numerical coefficients of o O 5x I o @ o o Simplify 5x G E - '11x. o Add the numerical coefficients of 0c the like variable terms. 6 (x+x)+(x+x+x) the like variable terms. 5x - 11x Lil-ri-"!r Ic o O 3:iffi,""0 -6x ln simplifying more complicated expressions, the following Properties of Addition are used. The Associative Property of Addition It a, b, and c are real numbers, then a When adding three or more 3x terms, the terms can be + 5x + b + c = (a+ b) + c : a *(b + c). * 9x = (3x + 5x) * 9x = 3x + (5x + 9x) 8x+9x-3x+14x grouped in any order. The sum is the same. 17x = 17x The Commutative Property ol Addition lf a and b are real numbers, then a * b = b + a. When adding two like terms, the terms can be added in either order. The sum is the same. 2x+(-4x)=-4x+2x -2x = -2x 62 UN lT 2 Variable Expressions The Addition Property ol Zerc : lf a is a real number, then a * Thesum of aterm andzero istheterm. 0 O +a= a. 5x + 0:0 * 5x =5x The lnverse Property of Addition lf a is a real number, then a + (-a) = (-a) The sum of a term and its opposite zero. The opposite of a number is called its additive inverse. Simplify 8x + 3y - is * a:0. 7x a (-7x) - -7x +7x -O 8x. Use the Commutative and Associative 8x + 3y - Bx Properties of Addition to rearrange r--------------r and group like combine tike terms. | 3y + (8x terms. - 8x) | Do these steps mentallv' |!l-t-t------l 3v= 3 E o Simplify 4x2 + 5x - 6x2 - 2x. E Use the Commutative and Associative 4x2 + 5x - 6x2 Properties of Addition to rearrange T-------t, and group like terms. @*, - 6rt) + (5x L-------- Combine like Example 1 Simplify 3x Solution 3x + 4y -7x + 11y Example3 Solution terms. a 4y -10x 10x + 7y +7y. x2-7 +4x2-16. x2 - 7 + 4x2 - 16 5x2 - 23 Simplify -2xz + Example2 E 2x - : Do these steps 2,-.-. € 6 1/-J mentaltY. 3x Simplify 3a -2b - 5a + 6b. Your solution Example4 Simplify -3y'+7 aByz -14. Your solution o $ !l d E o o c o 6 o UN lT 2 63 Variable Expressions 2.2 Objective To simplify a variable expression using the Properties of Multiplication ln simplifying variable expressions, the following Properties of Multiplication are USCd. The Associative Property of Multiplication lI a, b, and c are real numbers, then a. When multiplying three or more factors, the factors can be grouped in any order. The product is the same. b) . c = o. (b. c). 2(3x) : (2.3)x : 6x b. c = (a. The Commutative Properly ol Multiplication lf a and b are real numbers, then a . b = b. a. When multiplying two factors, the factors can be multiplied in either order. The product is the same. (2x) .3 = 3 . (2x) - (3 . 2)x - The Multiplication Property of One lf a is a real number, then a. 1 ='1 . e = d. term. o O The product of a term and one is the @ The lnverse Property of Multiplication (Bx)(1) r o o 6 E uo ! c 6 c o O lf a is a real number, and a is not equal to zero, then 1is = (1)(8x) = a': = :' a = Bx 1. I is atso called the multiplicative inverse of a, 7'| = l'l = I and its a'a called the reciprocal ol a. The product of a number reciprocal is one. Simplify 2(-x). of factors. Use the Associative Property Multiplication to group 2(-x) i----------r . | 2(-1 x) | Ll1_!-_1t_ll Do these steps mentailv -2x simplify;(?) Notethat +:?.i=3, ;(?) Multipli- ZG4 lni7;-l the i \Z'Sl^ Use the Associative Property of cation to group factors. Use the verse Property of Multiplication and Multiplication Property of One. 1x L--____J x i Do these steps mentally. 6x 64 UN lT 2 Variable Expressions Simplify (16x)2. Use the Commutative and Associative Properties of Multiplication to rearrange and group factors. (16x)2 T-------.1 i 2(16x) I L!1_L6I-, Oo these steps mentauy 32x 5 -2(3x2). Solution -2(3x2) Example Example Simplify 6 Simplify -5(4y2). Your solution -Gx2 Example 7 Example Simplify -5(-10x). Solution -5(-10x) 8 Simplify -7(-2a). Your solution 50x (6x)(-a). Solution (6x)(-4) Example 9 Example Simplify 10 Simplify (-5xX-2). Your solution 6 tt !i c E o 6 c o -24x !,o 2.3 Objective To simplify a variable expression using the Distributive Property A student works 3 hours on Friday and 5 hours on Saturday. The hourly rate of pay is $4 an hour. Find the total wages received for the two days. The total income can be found in two ways. 1. Multiply the hourly wage by the total number of hours 2. Find the income for each day and add. worked, $a(3) + $4(5) = $12 + $20 =932 = $4'8 =$32 that 4(3 + 5) = 4(3) + 4(5) This is an example of the Distributive Property $4(3 + 5) Note of Multiplication over Addition. The Distributive Property * c) : ab I ac (b+c)a-ba+ca. ll a, b, and c are real numbers, then a(b The Distributive Property is used to remove parentheses from a variable expression. 3(2x or - 5) -----------_.1 Do this steo Ltgl_*_:t_-_l_l il.rirru.'=" 6x - 15 oo I o @ o o o E Io !c co oo UN lT 2 Simplify Variable Expressions -3(5 + x). Use the Distributive Property to remove parentheses from the variabte -3(5 + x) expression. -' --'-j -15 Simplify - 3x - 4) Do this step I i-.(t;-t; L--'-'- mentally. -(2x - g. Use the Distributive Property to remove parentheses from the variabte (2x expression. i-i1;;: 4)-----l | -t (zx) - (- r )(4) I __ _______ ___l Do these sreps mentatty. L_ _ -2x+4 Notice: when a negative sign immediately precedes the parentheses, the sign ol each term inside the parentheses is changed. oo I o @ o 11 Simptify 7(4 + 2x). Solution 7(4 a 2x) Example 28 o + Exampte 12 Simptify 5(3 your solution + 7b). - t)5. 14x 6 E uo oc 6 C o o C) Example 13 Simplify (2x Solution (2x - 6)2 - 6)2. 4x-12 15 simplify -3(-5a + 7b). Solution -3(-5a 17b) 15a - 21b Exampfe 17 Simplify -(3a - 2). Solution -(3a - 2) Example Exampte 14 Simplify (3a your solution Exampte 16 simptify -8(-2a + Tb) Your solution Exampte 18 Simp[fy -(Sx your solution - t2). -3a+2 19 Simplify -2(xz + 5x - 4). Solution -2(xz + 5x - 4) -2x2 - 10x + B Example Example 20 Your sotution Simplify 3(-62 - 6a + 7). ro { rt ci E o o g o 6 6 66 UNIT 2.4 Objective 2 Variable Expressions I variable To simplify expressions When simplifying variable expressions, use the Distributive Property to remove parentheses and brackets used as grouping symbols. Simplify 4(x - y) - z(-gx + 6y). 4(x-y)-2(-3x+6y) 4x-4y*6x-12y Use the Distributive Property to remove parentheses. 'l0x Combine like terms. 21 - 3(2x Solution 2x - 3(2x - 7y) Exampfe Simplify 2x 7y). Example 22 Simplify 3y - - 1 6y 2(y - 7x), . Your solution 2x-Gx+21y -4x + 21y 23 Simplify 7(x - 2y) - 3(-x - 2y). Solution 7(x - 2y) - 3(-x - 2y) Example Example 24 Simplify -2(x - o 2y) + 4(x - o 3y). I o Your solution 7x-14y+3x+6y o 1Ox-By uo oc 6 E 6 C IC oo Example 25 Simplify -2(-3x q 7y) Solutlon -2(-3x + 7y) 6x-14y-14x -Bx Example 27 - 14x. 14x Example 26 Simplify -5(-2y - 3x) + 4y. Your solution 14y Simplify 2x-3[2x -3(x +7)]. Solution 2x - 3l2x - 3(x + 7)l ' 28 Example Simplify 3y - 2lx - aQ - 3y)1. Your solution 2x-312x-3x-211 2x-31-x-211 2x+3x+63 sx+63 6 rt r? .ri co 6 c e I oo UNIT 2 Vaiable Expressions 67 2.1 Exercises Simplify. 1. 6x+8x 2. 12x 4. 5. 4y 12a - 3a 7. -3b-7 10. -3a + + 11. 13. -12xy * Sab - 8y + (-oy) 9. -12a + 17a 3 12. 7ab 14. -1Sxy * 17xy 6. 1 (-10y) 8. -12y 12a 3. 9a-4a 13x 3xy - 9ab 3ab 15. -3ab + 3ab oo I o @ o o a 16. -7ab + 7ab 17. -L^ - le. tr" - #r, 20. 22. Bx+5x+7x 23.5a-3a+5a t, 18. -lv + f6t E Io !c d o c ()o 25. -5x2 - 12x2 + 3x2 26. -y2 - Bx 29. 28. By + (-10x) + 31. Sa + (-7b) - 5a + b 33. 3x + (-By) - 10x 35. x2 - 7x * + (-5x2) + 4x 5x 21.3x+5x+3x ?r - tr, 7x - 8yz 3y * + 7y2 10x 24. 10a 27. 7x + (-Bx) + 3y 17a + 30. By+8x-By 32. -5b + 7a - 7b 34. ' 3y + - (-12*) I 1Za 7, * ,, 36. 3x2+5x-10x2-lOx 3a .rj 68 UN lT 2 Variable Expressions 2.2 Exercises Simplify. 37. 4(3x) 38. 40. -2(5a) 41. -2(-3y) 42. -5(*6y) 43. (4x)2 44. 4s. 12(5x) (6x)12 39. -3(7a) (3a)(-2) 46. (7a)Ga) 47. (-3bx-4) 48. (-12bx-e) 49. 50. sr. ]{sr,) -5(3x2) -8(7y21 o s2. |{orr) s3. Ifsrl s4. *<trl x o @ 6 E - |r-zr7 s6. - I<-o"l 57. - |r-tn| qo E c 6 co c o o s8. - $r-so) 5e. 61. (-6y)(_ *) 62. (-1on)(-+) 64. (3D(+) |ttu) - |rr - f,{tza') - orl ${z+a") 70. -|t-avl 71. rs. (-6x)(+) 74. {-toxl(i) (16y)(+) 60. <tzxt($) 63. |rsrl - $rr o"l - |r-tavl 72. (ssy)(+) rs. (-8o(- q\ UN lT 2 Variable Expressions 69 2.3 Exercises Simplify. 76. -(x + 2) 77. -(x + 7) 78. 79. 5(2x 80. -2(a + 7) 81. -5(a + 16) 82. -3(2y 83. *5(3y - 84. 85. (1O 88. 3(5x2 - - 7) - 8) 7b)2 7) 2(4x (5 - - 3) 3b)7 86. -3(3 - 5x) 87. -5(7 - 89. 10x) 6(3x2 I 2x) 90. -2(-y + e) 91. -5(-2x + 7) 92. (-3x - 6)5 93. (-2x + 7)7 94. 2(-3yz - 14) 95. 5(-6x2 - 97. - 12) 100. -2(vz - 3y2) + 2x) I @ o @ 6 E r E c 6 c o o 103. -8(3y2 3(x2 + 2x - 6) 96. -3(Zyz 98. 3(x2:- yz1 99. s1x2 101. ,-4(vz - 5y') 102. -(6az - 7b2) 104. 105. -2(y' 2y + 4) 4(xz - 106. -3(y2 - 3y - 7) 107. Z(-sz 109. -5(-2xz - 3x + 7) 110. 112. 5(2yz - 4xy - y2) 3) 3x + 5) - 2a + 3) + - r, y2) - 108. 4(-3a2 - 5a + *3(-tvz + 3x - 4) 111. 3(2xz + xy - 113. -(3az + 5a - 4) 114. -(80" - 7,) 3y2) 6b + 9) UNIT 2 70 Variable Expressions 2.4 Exercises Simplify. 115. 4x - 2(3x + 116. 6a - (5a + B) 7) 117.9-3(4y+6) 118. 10 - (11x - 119. 5n - (7 - 12o. B-(12+4y) 2n) 121. 3(x + 2) - 5(x - 122. 2(x 7) 123. 12(y - 2) + 3(7 - 3y) - 4) - 3) 4(x + 2) 124, 6(2y - 7) - 3(3 - o 2y) I o @ o @ o 126. 125. 3(a - b) - 4(a + b) 2(a + 2b) (a - - E uo oc 3b) o c o co 127. 4Lx - 2(x - 128. Zlx + 2(x + 7)l 3)l 129. -213x + 2(4 - x)l 130. -512x + 3(5 - x)l 131. -312x - (x + 7)l 132. -2[3x 133. 2x - 3[x - 135. -5x:212x 2(4 - - (5x 134. -7x + 3[x - x)l 4(x + 7)l - -6 137. 2x +3(x -2y) + 5(3x -7y) - 2)] 7(3 - 2x)] 136. 4a - 2l2b - (b - 2a)l +eo 138. 5y -2(y - 3x) + 2(7x - y) UNIT 2 SECTION 3 3.1 Variable Expressions 71 Translating Verbal Expressions into Variable Expressions Obiective }rrffTt"te a verbal expression into a variable expression given the One of the major skills required in applied mathematics is to translate a verbal expres- sion into a variable expression. This requires recognizing the verbal phrases that translate into mathematical operations. A partial list of the verbal phrases used to indicate the different mathematical operations is given below. Addition 6 added to y 8 more than x the sum of x and z I increased by 9 the total of 5 and y added to more than the sum of increased by the total of o o Subtraction minus r less than a decreased by o the difference between 6 E I! c o c o Multiplication o Division Power Translate Step 1 Step 2 v +6 X+B X+Z 1+g 5+y xminus2 x-2 Tlessthanf t-7 mdecreasedby3 m-3 thedifferencebetween yand4 y 4 - times 10 times f 101 of one half of x Lr the product of multiplied by the product ol y and z y multiplied by 11 yz divided by x divided by n the quotient of the quotient ot y and z 11v X 12 v z the ratio of the ratio of t to 9 I the square of the cube of the square of x the cube of a x. "l4less than the cube of x" t a3 into a variable expression. ldentify the words which indicate the mathematical operations. 14 less than the cube of x Use the identified operations to x3 write the variable expression, - 14 UNIT 72 2 Variable Expressions Translate "the difference between the square of x and the sum of y and variable expression, Example 1 into a Step 1 ldentify the words which indicate the difference between the square the mathematical operations. of x and the sum of y and z Step 2 Use the identiiied operations write the variable expression, Translate "the total of 3 times n and n" into a variable expres- to x2 Example 2 sion. Solution z" the total of 3 times n and n - (y + z) Translate "the difference between twice n and one third of n" into a variable expression. Your solution 3n+n Example 3 Translate "m decreased by the Example 4 sum of n and 12" into avariable expression. Solution m decreased by the sum of n Translate "the quotient ol 7 less than b and 15" into a variable expression. Your solution and 12 m-(n+12) @ tt tl ci c o tt E o oo o O I o 3.2 Obiective 9a il.",#lr:?te a verbal expression into a variable expression by assigning o 6 E I c ln most applications which involve translating phrases into variable expressions, the oc variable to be used is not given. To translate these phrases, a variable must be as- co o signed to an unknown quantity before the variable expression can be written. @ Translate "the sum of two consecutive integers" into a variable expression. Step 1 Assign a variable to one the unknown quantities, of Step 2 Use the assigned variable to write an expression for any other unknown quantity the next consecutive integer: n Step 3 Use the assigned variable to write the variable expression. n + (n + the first integer: n 1) + 1 UN lT 2 73 Variable Expressions Translate "the quotient of twice a number and the difference between the number and twelve" into a variable expression. Example 5 of Step 1 Assign a variable to one the unknown quantities. Step 2 Use the assigned variable to write an expression for any other unknown quantity. Step 3 Use the assigned variable to write the variable expression. "a number added to the product of four and the square of the number" into a Translate the unknown number: n twice a number: 2n the difference between the number and twelve: n - 12 2n J Example 6 "a number multiplied by the iotal of len and the cube of the number" into a variable Translate expression. variable expression. Solution the unknown number: n Your solution the square of the number: n2 the product of 4 and the square oJ the number: 4n2 o O n+4nz I o @ o o o E o L ! c Example 7 c o o O Translate "the sum of thesquares of two consecutive integers" into a variable expres- Example 8 Translate "the sum of three consecutive integers" into a variable expression' sion. Solution the Jirst integer: x the second integer: Your solution x1 1 x2y(xa1)2 Example ' 9 Translate "four times the sum of one half of a number and fourteen" into a variable ex- Example 10 pression. Solution the unknown number: n one half ol the number, Translate "five times the difJerence between a number and sixty" into a variable expression. Your solution ln the sum of one half of the number and fourteen: I 2n+14 +(ln + t+) (o rt s .i c o o q ! o o UNIT 74 Variable ExPressions 2 ffiariable a verbal expressron exPression and 1: . To translate e{Egslg! resulting the then simPlifY 3.3 Obiective Aft e r transr ati n s a ve rbar j' F io* n'"' "[, J,?,H, flJ,ilf::il-Siff il:llu :iffi::' l?:ilil'xHr[i'::":1:::::x::T:i:: ice the number and lff Lx3:lii'J^f between lwice dirference the "a number minus *o*u"i3;fi ;t#;i n eleven." I SteP 1 vivP the unknown number: one of Assign a variable to tne Jnxnown quantities' twice the difference between '"inl'n-urno.r and eleven: 2n *igblt^l: SteP 2 Use the assigned {or anY write an exPression other unknown quantitY' Step 3 Step Example 11 4 Solution - 2n + 11 Simpllly the variable n expression. -n + 11 Example of one fourth of a number and one eighth of the same number." the unknown number: n 11 n-(2n-11) Use the assigned variable to wrrte the variab\e exPression. Translate and simplify "the sum = 12 Translate and simplify ,,the difference between three eighths of a number and five sixths of the same number." o o f o o o @ 6 E Your solulion Io c c o c one fourth of the number, o !n one eighth of the number, |n .a ln+ln f,n+ln "61 U 14 Example 13 Translate and simplify "the total of five times an unknown number and twice the difference between the number and nine." Example Solution the unknown number: n five times the unknown number: Your solution Translate and simplify ,,the sum of three consecutive integers.,' 5n twice the difference between the number and nine: 2(n - 9) 5n+2(n-9) 5n+2n-18 7n-18 @ rt a A c o , E o .:o ah UN lT 2 75 Variable Expressions 3.1 Exercises Translate into a variable expression. y 2. a less than 16 increased by 10 4. p decreased by added 1o 14 6. q multiplied by 13 8. 6 times the difference between m 1. the sum of 8 and 3. I 5. z 7. 20 less than the square of x 7 and 7 9. o O 11. the sum of three fourths of n 12 and 10. b decreased by the product of 2 and b B increased by the quotient of n 12. y 14. 8 the product of and 4 T o @ o -B and y (, 6 E o I ! 13. the product of 3 and the total of and C 7 c o o O 15. the product of f and the sum of and 16 f 17. 1 5 more than one half of the square of 19. divided by the difference bex and 6 tween x 16. 18. the quotient of 6 less than n and twice n 1 9 less than the product of n and -2 the total of 5 times the cube of n 20. the ratio of 9 more than m to m and the square of n 21. r decreased by the quotient of r 22. four fifths of the sum of w and 10 and 3 23. the difference between the square 24. s increased by the quotient of 4 of x and the total of 25. x and 1 7 the product of 9 and the total of and 4 and s z 26. n increased by the difference between 10 times n and 9 76 UN lT 2 Variable Expressions 3.2 Exercises Translate into a variable expression. 27. twelve minus a number 28. a number divided by eighteen 29. two thirds of a number 30. twenty more than a number 31. the quotient of twice a number and nine 32. ten times the difference between a 33. eight less than the product number and fifty of 34. the sum of five eighths of a number and six nine less than the total of a number and two 36. the product of a number and three more than the number eleven and a number 35. ,a I o @ o 37. the quotient of seven and the total of five and a number 38. four times the sum of a number and nineteen o o E Io c o c o E o 39. five increased by one half of the sum of a number and three 40. 41. a number multiplied by the difference between twice the number and four 42. the product 43. the product of five less than a the quotient of fifteen and the sum of a number and twelve of two thirds and the sum of a number and seven 44. the difference between forty and the quotient of number and the number a number and twenty 45. the quotient of five more than twice a number and the number 46. the sum of the square of a number and twice the number 47. a number decreased by the differ- 48. the sum of eight more than a number and one third of the number ence between three times the number and eight UN lT 2 77 Variable Expressrons 3.3 Exercises Translate into a variable expression. Then simplify. 49. 51. a number added 1o the product three and the number five more than the sum of a ber and six of 50. a number increased by the total of the number and nine num- 52. a number decreased by the difference between eight and the number 53. a number minus the sum of number and ten the 54. the difference between one third of a number and five eighths of the number 55. o O the sum of one sixth of a number and four ninths of the number 56. two more than the total of a number and five 58. twice the sum of six times a number and seven seventh 60. Jour times the product of six and a I o o o E uo 57. the sum of a number divided by three and the ! c o c number C o O 59. fourteen multiplied by one of a number number 61. the difference between ten times a number and twice the number 62. the total of twelve times a number 63. sixteen more than the difference between a number and 64. a number plus the product of the number and nineteen number subtracted from the product of the number and four 66. eight times the sum of the square of a number and three six 65. a and twice the number fifteen 68. two thirds of the sum of nine times a number and three times a number and the product of 67. the difference between the number and five 78 UN lT 2 Variable Expressions Translate into a variable expression. Then simplify. 69. twelve less than the difference between nine and a number 70. thirteen decreased by the sum of a number and fifteen 71. ten minus the sum of two thirds of a number and four 72- nine more than the quotient of eight times a number and four 73. five times the sum of two consecu- 74. seven times the total of two consecutive integers 76. five minus the sum of two consecutive integers tive integers 75. six more than the sum of two consecutive integers oo I @ 77. twice the sum of three conseculive integers 78. o one third of the sum of three con- o secutive integers E o L ! c o c o c o O 79. the sum of three more than the square of a number and twice the square of the number 80. the total of five increased by the cube of a number and twice the cube of the number 81. a number plus seven added to the difference between two and twice 82. ten more than a number added to the difference between the number and eleven the number 83. six increased by a number added to twice the difference 84. the sum of a added between plus the product of number minus nine and four between the number and twelve the number and three 85. a number number and ten to the difference a 86. eighteen minus the product of two less than a number and eight UNIT 2 Variable Expressions Review SECTION SECTION 1 2 1.1a 2.1a 79 /Test Evaluate b2 - 3ab when €=3andb:*2. Simplify 3x 2.1c Simplify 3x 1.1b Evaluate #whena - -4 andb=6. - 5x + 7x 2.1b - 7y - 12x. 2.1d Simplify -7y, + 6y, - (-2y2). Simplify 3x*(-12y)-5x-(-7y). oo I @ o o E r ! c c o c o Simplify f {t orl. 2.2c Simplify (-g)(-12D. 2.3a Simplify 5(3 o - 7b). 2.2b 2.3b simplify (12D(!) Simplify f t-r srl. Simplify -2(2x - $. 80 UN lT 2 Variable Expressions Review/Test 2.3c Simplify 2.4a -3(214 Simplify 2x - - 3(x 7y,) - 2). 2.4c Simplify 2x+314-(3x-7)1. 2.3d Simplify -5(2sP 2.4b Simplify 5(2x 2.4d Simplify - 3x + 6). + 4) - 3(x - 6). -2lx-2(x-D)+sy. o O r SECTION 3 3.1a Translate "b decreased by the product of b and 7" into a varia- 3.1b Translate "10 ence between o @ times the differ- e x o and 3" IOIO d 5 E variable expression. ble expression. Io .! c o zo E oo 3,2a Translate "the sum of a number and twice the square of the number" into a variable expression. 3.3a Translate and simplify "eight times the sum of two consecutive integers. " 3.2b Translate "three less than the quotient of six and a number" into a variable expression. 3.3b Translate and simplify "eleven added to twice the sum of a num- ber and four." UNIT 2 Variable Expressions 81 Review/Test SECTION 1 1.1a . SECTION 2 2.1a Evaluate a2 - 3b when a=2andb=-4. a)-4 b) 16 c)4 d) 14 Simplify 3x a) -21x b) 17x c) 7x d) -17 x - 8x + (*12x). 2.1c Simplify 5a - 1Ob a) -27 a b) 5a - 22b c) -7a - 10b d) -5b - 12a o o I o @ o o a 12a. 1.1b Evaluate 3(a c) - - 2ab when a=2, b:3, andc= -4. a)6 b) -6 c) 12 d) -12 2.1b Simplify a) b) c) d) -2x2 5x2 - Ge^r1 + 9x2 6x2 -x2 2.1d Simplify 2a - (-3b) a) 5a - 12b b) -5a - 2b c) 5a-2b d) -5a + 8O 2.2b simplify (16r)(+) E o I !c g o o O 2.2a Simplify lfiz"1. a) 24a b) -24a c) -6a d) 6a 2.2c Simplify -4(-gy). 2.2d 128x Bx 4x 2x simprify (-i)f a) 36y b) -36y a) -30b b) 30b c) 25b d) iv d) c) 2.3a a) b) c) d) 49y Simplify 3(B - d) -24 - 6x a) 6x-24 b) -24 + 6x c) 24-Gx 2x). 2.3b 4x2. -soo; 56b Simplify -2(-3y + 9). a) -6Y + tg b) -Gy - 18 c) 6Y+te d) 6y-18 7a - 5b. UN 82 lT 2 Variable Expressions Review/Test 2.3c Simplify -4(2x2 - a) b) c) d) 8x2 - 12Yz -8x2 + -Bxz Bxz Simplify a - 12y2 3Y2 -3x - - 3Y a) -9y' + 9y + 21 b) -9Y' - 9Y - 21 c) -9Y" -3Y -7 7). SimplifY -3(3Y2 d) 9Y2-3Y-7 3y2 2(2x a) x - 14 b) -7x + 4 c) -7x + 14 d) -x - 7 2.4c 2.3d 3Y')' - Simplify 4(3x 7). a) 5x+27 b) 19x - 27 c) 19x - 3 d) 5x-43 2.4d Simplify zx+3lx-2(a-2x)). a) 17x - 24 b) 15x - 22 c) 17x-B d) 9x - 10 - 2) - 7(x + 5) Simplify - 3(x - 2Y)) + 3Y a) -3x + 211 b) 3x+6y c) -3x t 18y d) 9x .1 3y 3[2x ' o O r @ SECTION 3 3.1a 3.2a Translate "the sum o{ one half of b and b" into a variable exPres- divided by the difTranslate between ference Y and 2" into a sion. variable exPression. a) (]o)rot b) f,fo + o1 c) L*, d) f,o+a a) Translate b) c) d) "the difference be- tween eight and the quotient of a number and twelve" into a varia- ble expression. a) 8-+ c) 8-+ 3.3a 3.1b "l0 b) "# d) 8-12n Translate and simPlifY "the Product of four and the sum of two consecutive i ntegers. " a) b) c) d) 5+2x 8x*4 6+2x Bx+1 o oc o E 10 c o O 1ov 2 1o-v 2 10 2y Translate "the Product of a number and two more than the num- ber" into a variable exPression. a) 2n -12 b) n(n + 2) c) n+n+2 d) 2n(n) "twelve more than the Product of three plus a number and iive." a) 5x + 5'l b) 3x+27 c) 20+x d) 5x+27 E I y-t 3.3b Translate and simplify o

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