Transcript Practice Test - USF Math Lab

Section 1.1
Numbers and Their
Properties
OBJECTIVES
A
Write a set of numbers
using roster or set–
builder notation.
OBJECTIVES
B
Write a rational number
as a decimal.
OBJECTIVES
C
Classify a number as
natural, whole, integer,
rational, irrational, or
real.
OBJECTIVES
D
Find the additive inverse
of a number.
OBJECTIVES
E
Find the absolute value
of a number.
OBJECTIVES
F
Given two numbers, use
the correct notation to
indicate equality or
which is larger.
DEFINITION
NATURAL NUMBERS
The set of numbers used for
counting.
N = {1, 2, 3, . . . }
DEFINITION
WHOLE NUMBERS
The set of natural numbers
and zero.
W = { 0, 1, 2, 3, . . .}
DEFINITION
INTEGERS
The set of whole numbers and
their opposites(negatives).
I = {. . . , –2, –1, 0, 1, 2, . . .}
DEFINITION
RATIONAL NUMBERS
All numbers that can be written
as the ratio of two integers.
a
Q = {r |r = , a and b integers, b  0 }
b
DEFINITION
IRRATIONAL NUMBERS
Numbers that cannot be written
as ratios of two integers.
H = {x | x is a number that is not rational}
DEFINITION
REAL NUMBERS
Numbers that are either
rational or irrational:
R = {x | x a number that is rational or irrational}
DEFINITION
ADDITIVE INVERSE
The additive inverse(opposite)
of a is –a.
DEFINITION
ABSOLUTE VALUE
The distance between a and
0 on the real-number line
|a | =
a if a is positive |11| = 11
0 if a is zero
|0 | = 0
-a if a is negative |-5 | = - (-5) = 5
CAUTION
The absolute value is always
positive or zero.
–|– 3| = –3,
–|4.2| = –4.2
DEFINITION
TRICHOTOMY LAW
If given any two real numbers,
only one of three things is true:
1. a is equal to b, denoted by a = b, or
2. a is less than b, denoted by a < b, or
3. a is greater than b, denoted by a >b.
Chapter 1
The Real Numbers
Section 1.1A
Practice Test
Exercise #1
Use roster notation to list the natural
numbers between 5 and 9.
The set of natural numbers
between 5 and 9 is {6, 7,8}
Note 5 and 9 are not included
Chapter 1
The Real Numbers
Section 1.1B
Practice Test
Exercise #2
Write as a decimal:
3
a. = 8 3
8
2
b. = 3 2
3
= 8 3.000
= 3 2.000
= 0.375
= 0.666...
= 0.6
Chapter 1
The Real Numbers
Section 1.1C
Practice Test
Exercise #3
Classify the given number by making a
check mark () in the appropriate row(s).
Set
Natural number
Whole number
Integer
Rational number
Irrational number
Real number
0.5 0 – 6
2
7
5

 
   

    
Chapter 1
The Real Numbers
Section 1.1D
Practice Test
Exercise #4
4
Find the additive inverse of .
5
4
=–
5
Chapter 1
The Real Numbers
Section 1.1E
Practice Test
Exercise #5
Find:
a. | –9 | = 9
b. | 0.5 | = 0.5
Chapter 1
The Real Numbers
1.1F
Practice Test
Exercise #6
Fill in the blank with <, >, or = to
make the resulting statement true:
1
1
a. – _____
> –
4
3
1
1
so
–
= – 0.25
= 4 1 = 4 1.00 = 0.25
4
4
1
1
= 3 1 = 3 1.00 = 0.66 so – = – 0.66
3
3
– 0.66 is farther from 0 than – 0.25
so – 0.25 is greater than – 0.66
2
=
b. 0.4 _____
5
2
=5 2
5
= 5 2.0
= 0.4
Section 1.2
Operations and
Properties of Real
Numbers
OBJECTIVES
A
Add, subtract, multiply,
and divide signed
numbers.
OBJECTIVES
B
Identify uses of the
properties of the real
numbers.
PROCEDURE
TO ADD TWO NUMBERS
WITH THE SAME SIGN:
Add their absolute values and
give the sum the common
sign.
PROCEDURE
TO ADD TWO NUMBERS
WITH DIFFERENT SIGNS:
1. Find the absolute value.
2. Subtract the smaller from the
greater number.
3. Use the sign of the number with
the greater absolute value.
DEFINITION
ADDITIVE IDENTITY
For any real number a:
a +0 =a =0 + a
DEFINITION
SUBTRACTION OF SIGNED
NUMBERS
If a and b are real numbers:
a - b = a + (-b)
DEFINITION
ADDITIVE INVERSE
For any real number a:
a + (-a) = (-a) + a = 0
PROCEDURE
SIGNIFY MULTIPLICATION
raised dot : a • b
next to each other : ab
parentheses: (a)(b), a(b), or (a)b
PROCEDURE
MULTIPLYING NUMBERS WITH
OPPOSITE SIGNS
To multiply a positive number
by a negative number, multiply
their absolute values and make
the product negative.
DEFINITION
SIGNS OF MULTIPLICATION
PRODUCTS
Same signs: Positive(+)
Different signs: Negative(–)
DEFINITION
IDENTITY FOR
MULTIPLICATION
For any real number a:
a • 1 =1 • a = a
DEFINITION
MULTIPLICATION OF
FRACTIONS
a
b
•
c=a
d b
•
•
c (b, d  0)
d
DEFINITION
DIVISION OF REAL NUMBERS
If a and b are real numbers
and b is not zero:
a = q means that a = b • q
b
DEFINITION
SIGNS OF A FRACTION
For any real number a and
nonzero real number b, there
are two cases of signs:
-a = a = - a or -a = a
b -b b
-b b
DEFINITION
ZERO IN DIVISION
For a ≠ 0:
0 = 0 and a = undefined
a
0
CAUTION
0 is okay but n is a no-no!
k
0
DEFINITION
MULTIPLICATIVE INVERSE
(RECIPROCAL)
Every nonzero real number a
has a reciprocal such that:
1
•
a a =1
DEFINITION
DIVISION OF FRACTIONS
a ÷ c=a
b d b
•
d (b, c and d  0)
c
Chapter 1
The Real Numbers
Section 1.2A
Practice Test
Exercise #7
Find.
a. –9 + 5
= –4
b. –0.8 +  –0.7 
= –1.5
Chapter 1
The Real Numbers
Section 1.2A
Practice Test
Exercise #8
Find.
a. –16 – 7
=  –16  +  –7 
= –23
b. – 0.6 –  –0.4 
= –0.6 + 0.4
= –0.2
Chapter 1
The Real Numbers
Section 1.2A
Practice Test
Exercise #9
Find.
1 3
a. – –
8 4
=
   
1
3
–
+ –
8
4
Least common denominator = 8.
=
   
–
1
6
+ –
8
8
Now add numerators.
=
 –1 +  –6 
–7
=
8
7
= –
8
8
Find.
 
3
5
=– + 
4
6
b. –
3
5
– –
4
6
Least common denominator = 12.
9 10
= – +
12 12
Now add numerators.
–9 +10
=
12
1
=
12
Chapter 1
The Real Numbers
Section 1.2A
Practice Test
Exercise #10
Find.
a. 6  –9 
= – 6 9
= –54
b. –4  –1.2 
= +  4 12 
= 4.8
Chapter 1
The Real Numbers
Section 1.2A
Practice Test
Exercise #11
Find.
1
a. –
2
2
9
1
= –
1
9
1
Find.
3 9
b. – ÷
2 8
3
= –
2
4
= –
3
1
1
8
9
4
3
Chapter 1
The Real Numbers
Section 1.2B
Practice Test
Exercise #12
Name the property illustrated in the statement.
a.  7 + 3  + 6 =  3 + 7  + 6
Commutative Property of Addition
b.  2 + 9  + 4 = 2 +  9 + 4 
Associative Property of Addition
Chapter 1
The Real Numbers
Section 1.2B
Practice Test
Exercise #13
Name the property illustrated in the statement.
1
a. 3
=1
3
Inverse Property of Multiplication.
b. 0.3 +  –0.3  = 0
Inverse Property of Addition.
Section 1.3
Properties of
Exponents
OBJECTIVES
A
Evaluate expressions
containing natural
numbers as exponents.
OBJECTIVES
B
Write an expression
containing negative
exponents as a fraction.
OBJECTIVES
C
Multiply and divide
expressions containing
exponents.
OBJECTIVES
D
Raise a power to a
power.
OBJECTIVES
E
Raise a quotient to a
power.
OBJECTIVES
F
Convert between
ordinary decimal
notation and scientific
notation, and use
scientific notation in
computations.
DEFINITION
EXPONENT AND BASE
If a is a real number and n is
a natural number:
an = a • a • a • • • a
n factors
Chapter 1
The Real Numbers
Section 1.3A
Practice Test
Exercise #14
Evaluate.
a.
 –3 4 =  –3 –3 –3 –3
= +9+9
= 81
b.
–34 = –  3 3 3 3 
= – 81
Chapter 1
The Real Numbers
Section 1.3B
Practice Test
Exercise #15
Write without negative exponents.
a. 7 –2 = 1 = 1 = 1
72 7 7 49
b. x –8 = 1
x8
Chapter 1
The Real Numbers
Section 1.3C
Practice Test
Exercise #16
Perform the indicated operation and simplify.
a. (3x 4y)(–4x –8y 8)
48x 4
b.
16x –8
Perform the indicated operation and simplify.
a. (3x 4y)(–4x –8y 8)
= 3  –4  x 4 x –8 y 1 y 8
= –12 x 4 +
 –8 
y1 + 8
= –12x –4y 9
= –12
1
x4
y9
–12y 9
12y 9
=
= –
4
x
x4
Perform the indicated operation and simplify.
48x 4
b.
16x –8
48
=
16
= 3 x
x4
x –8
4 –  –8
= 3x 4 + 8
= 3x 12
Chapter 1
The Real Numbers
Section 1.3D, E
Practice Test
Exercise #17
Simplify.
a. (–2x 8y –2)3
= (–2)3 (x 8)3 (y –2)3
 
=  –2  –2  –2  x 8 3y –2
= –8x 24 y –6
–8x 24
=
y6
8x 24
= –
y6
3
Simplify.
5
 x 
b.  –3 
y 
–3
(x5) –3
=
(y –3) –3
x 5(–3)
=
y (–3)(–3)
x –15
1
=
=
9
y
x 15
1
1 =
x 15y 9
y9
Chapter 1
The Real Numbers
Section 1.3F
Practice Test
Exercise #18
Write in standard notation.
6.5 x 10–3
The exponent of 10, (–3), means
move the decimal point 3
places to the left.
= .006.5
= 0.0065
Chapter 1
The Real Numbers
Section 1.3F
Practice Test
Exercise #19
Write as a whole number.
8.5 x 105
The exponent of 10, (5), means
move the decimal point 5
places to the right.
= 8.50000
= 850,000
Chapter 1
The Real Numbers
Section 1.3F
Practice Test
Exercise #20
Perform the calculation and write your answer in
scientific notation.
 7.1  105    4  10 – 7 
=  7.1  4    105  10–7 
= 28.4  105 + (–7)
=  28.4   10–2
NOTE
28.4 = 2.84  101
Perform the calculation and write your answer in
scientific notation.
 7.1  105    4  10 – 7 
=  28.4   10–2
=  2.84  101  10–2
= 2.84  101 + (–2)
= 2.84  10–1
Section 1.4
Algebraic
Expressions and
The Order of
Operations
OBJECTIVES
A
Evaluate numerical
expressions with
grouping symbols.
OBJECTIVES
B
Evaluate expressions
using the correct order
of operations.
OBJECTIVES
C
Evaluate algebraic
expressions.
OBJECTIVES
D
Use the distributive
property to simplify
expressions.
OBJECTIVES
E
Simplify expressions by
combining like terms.
OBJECTIVES
F
Simplify expressions by
removing grouping
symbols and combining
like terms.
PROCEDURE
ORDER OF OPERATIONS
P
E
1. Do the operations in the ().
2. Evaluate exponential
expressions.
(MD) 3. Perform multiplications and
divisions from left to right.
(AS) 4. Perform additions and
subtractions from left to right.
PROCEDURE
Identity for Multiplication
For any real number a:
a =1•a
PROCEDURE
Additive Inverse
For any real number a:
– a = –1 • a
PROCEDURE
Additive Inverse of a Sum
–(a + b) = –a – b
PROCEDURE
Additive Inverse of a
Difference
–(a – b) = –a + b
DEFINITION
LIKE TERMS
Constant terms or terms with
exactly the same variable
factors are called similar or
like terms.
Chapter 1
The Real Numbers
Section 1.4A
Practice Test
Exercise #21
Evaluate.
a. [ –7(4 + 3)] + 9 = –7(7) + 9
= –49 + 9
= –40
11
5 (131 – 32) 5(99)
b.
=
9
1 9
= 55
Chapter 1
The Real Numbers
Section 1.4B
Practice Test
Exercise #22
Evaluate.
6 – 12
3
–4 +
+ 15 ÷ 3
2
6 – 12
+ 15 ÷ 3
2
–6
= – 64 +
+ 15 ÷ 3
2
= – 64 +
= – 64 – 3 + 5
= –62
Chapter 1
The Real Numbers
Section 1.4C
Practice Test
Exercise #23
Evaluate.
a. 1 (b1 + b2)h gives the area of a
2
trapezoid. Find the area of the
trapezoid if b1 = 8, b2 = 3, and h = 6.
1
= 8 + 3 6
2
1 116
=
2
= 33
Evaluate.
b. 7 – x 2 + 20 ÷ y  – xy;
if x = –2 and y = 4.
= 7 – (–2) 2 + (20 ÷ 4) – (–2 4)
= 7 – (–2) 2 + 5 – (–8)
= 7–4+5+8
= 16
Chapter 1
The Real Numbers
Section 1.4D, E
Practice Test
Exercise #24
Simplify.
a. –5  x + 7 = –5x – 35
b. 7x – 3x + 1 + 2x + 2
= 7x – 3x – 1 + 2x + 2
= 6x + 1
Chapter 1
The Real Numbers
Section 1.4F
Practice Test
Exercise #25
Simplify.
[(5x 2 – 3) + (3x + 7)] – [(x – 3) + (2x 2 – 2)]
=
5x 2 – 3 + 3x + 7 – x – 3 + 2x 2 – 2

 
= 5x 2 + 3x + 4 – 2x 2 + x – 5
2
2
= 5x + 3x + 4 – 2x – x + 5
= 3x 2 + 2x + 9
