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```UNIT
2
i+ ,/
r.
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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.
2a2
-5a
Example
3
Sofution
Evaluate x2
-
Example
3xy when
X=3andY=-4.
4
Evaluate 2xy
+
y2 when
X=-4andy-2.
- 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
++
_
(_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.
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
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
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
o
O
5x
I
o
@
o
o
Simplify 5x
G
E
-
'11x.
o
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.
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
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
ol Zerc
:
lf a is a real number, then a
*
Thesum of aterm andzero
istheterm.
0
O
+a=
a.
5x +
0:0 *
5x =5x
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
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.
Example4
Simplify
-3y'+7
aByz
-14.
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).
-Gx2
Example
7
Example
Simplify
-5(-10x).
Solution -5(-10x)
8
Simplify
-7(-2a).
50x
(6x)(-a).
Solution (6x)(-4)
Example
9
Example
Simplify
10 Simplify (-5xX-2).
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
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
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
+
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
Exampte
16 simptify -8(-2a
+
Tb)
Exampte
18
Simp[fy -(Sx
-
t2).
-3a+2
19 Simplify -2(xz + 5x - 4).
Solution -2(xz + 5x - 4)
-2x2 - 10x + B
Example
Example
20
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),
.
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
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.
14y
Simplify
2x-3[2x -3(x +7)].
Solution 2x - 3l2x - 3(x + 7)l
' 28
Example
Simplify
3y
-
2lx
- aQ -
3y)1.
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.
8 more than x
the sum of x and z
I increased by 9
the total of 5 and y
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.
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.
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.
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
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:
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.
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
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:
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
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
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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