Download Math 101 Study Session Quiz 2 Chapter 5 Sections 1 through 3

Math 101 Study Session Quiz 2 Chapter 5 Sections 1
through 3
July 28, 2016
Chapter 5 Section 1: Evaluating Variable Expressions
A variable is a letter of the alphabet that is used to stand for a quantity that is unknown
or that can change or vary.
An expression that contains one or more variables is called a variable expression.
Note: terms in an expression are separated by plus signs (+) and minus signs (−) that are
NOT contained in grouping symbols.
Terms that contain variables are called variable terms. Terms that do not contain variables
are called constant terms or simply constants.
Replacing each variable in a variable expression by its value and then simplifying the resulting
numerical expression is called evaluating a variable expression.
Chapter 5 Section 2 Simplifying Variable Expressions
Like terms of a variable expression are terms with the same variable part.
The Distributive Property
If a, b, and c are real numbers, then a(b + c) = ab + ac or (b + c)a = ba + ca.
If
If
If
If
If
If
The Associative Property of Addition
a, b, and c are real numbers, then (a + b) + c = a + (b + c).
The Commutative Property of Addition
a and b are real numbers then a + b = b + a.
The Addition Property of Zero
a is a real number, then a + 0 = a and 0 + a = a.
The Inverse Property of Addition
a is areal number, then a + (−a) = 0 and (−a) + a = 0.
The Associative Property of Multiplication
a, b, and c are real numbers, then (ab)c = a(bc).
The Commutative Property of Multiplication
a and b are real numbers, then ab = ba.
1
The Multiplication Property of One
If a is a real number, then a · 1 = a and 1 · a = a.
The Inverse Property of Multiplication If a is a real number and a is not equal to
1
1
1
1
is called the reciprocal of a.
is also called the
zero, then a · = 1 and · a = 1.
a
a
a
a
multiplicative inverse of a. The product of a number and its reciprocal is 1.
Chapter 5 Section 3 Translating Verbal Expressions into
Variable Expressions
Words or Phrases for Addition
added to
6 added to y
y+6
more than
8 more than x
x+8
the sum of
the sum of x and z x + z
increased by
t increased by 9
t+9
the total of
the total of 5 and d 5 + d
plus
b plus 17
b + 17
Words or Phrases for Subtraction
minus
x minus 2
less than
7 less than t
less
7 less t
subtracted from
5 subtracted from d
decreased by
m decreased by 3
the difference between
the difference between y and 4
Words or Phrases for Multiplication
times
of
the product of
multiplied by
twice
x−2
t−7
7−t
d−5
m−3
y−4
10 times t
10t
1
one-half of x
x
2
the product of x and y xy
b multiplied by 11
11b
twice n
2n
Phrases for Division
divided by
x divided by 12
the quotient of
the quotient of y and z
the ratio of
the ratio of t to 9
x
12
y
z
t
9
2
Phrases for Power
the square of
the square of x
the cube of
the cube of a
x2
a3
Below are some examples for us to try with solutions at the end of the study
guide:
Solve each of the following:
1. Simplify
2
3
11
− a− − a − a
5
10
15
2. Simplify
3
(6x − 10y + 12)
4
3. Simplify
−7 [4x + 2(7 − x)]
4. Evaluate the variable expression when a = −2, b = 6, c = −5, and d = 4.
1
−5
b + (ac + bd)
6
2
5. Translate the verbal expression “the sum of one-sixth of a number and four-fifths of
the number” into a variable expression using the variable x to represent number. Then
simplify the variable expression.
3
Solutions to the Study Guide Questions
Solution to 1:
The LCD of 5, 10, and 15 is 30.
2
3
11
2
3
11
− a− − a − a=− a+ a− a
5
10
15
5
10
15
2 6
3 3
11 2
=− · a+
· a−
· a
5 6
10 3
15 2
12
9
22
=− a+ a− a
30
30
30
−12 + 9 − 22
=
a
30
−34 + 9
=
a
30
−25
=
a
30
−5
=
a
6
−5a
=
6
Solution to 2:
3
3
3
3
(6x − 10y + 12) = (6x) + (−10y) + (12)
4
4
4
4
3 · 6x 3 · −10y 3 · 12
=
+
+
4
4
4
18x −30y 36
=
+
+
4
4
4
9x 15y
=
−
+9
2
2
Solution to 3:
−7[4x + 2(7 − x)] = −7[4x + 2(7) + 2(−x)]
= −7[4x + 14 − 2x]
= −7(4x) + −7(14) + −7(−2x)
= −28x − 98 + 14x
= −14x − 98
Solution to 4:
To evaluate the variable expression, replace each a with -2, each b with 6, each c with -5,
and each d with 4.
4
1
−5
1
−5
b + (ac + bd) =
· (6) + ((−2)(−5) + (6)(4))
6
2
6
2
−30 1
=
+ (10 + 24)
6
2
1
= −5 + (34)
2
34
= −5 +
2
= −5 + 17
= 12
Solution to 5:
Variable Expression:
1
4
x+ x
6
5
Simplified Variable Expression:
1
4
5 1
64
x+ x= · x+
x
6
5
5 6
65
5
24
= x+ x
30
30
29
= x
30
Thus, the simplified variable expression is
29
x.
30
Questions for You to Try
Below are some questions for you to try on your own.
1. Evaluate the variable expression when a = −2, b = 4, c = −2, and d = 3.
2
1
− d − (bd − ac)
3
4
2. Simplify
x2 − 6x + (−3x2 ) + 3x
3. Simplify
−6(−9x2 + 6x − 9)
4. Simplify
2x − 9[x − (2 − x)]
5
5. Translate the verbal expression “the quotient of five more than twice a number and
the number” into a variable expression using x to represent number. Then simplify the
variable expression.
Solutions
1. -4
2. −2x2 − 3x
3. 54x2 − 36x + 54
4. −16x + 18
5. Variable Expression:
2x + 5
5
Simplified Variable Expression: 2 +
x
x
6