Chapter 1: Sets, Operations and Algebraic Language
1.4: Operations with Signed Numbers (Including Absolute Value)
Operations with Signed Numbers
Operation
Same Signs
Addition
ADD and KEEP the sign.
Multiplication
and Division
Product or Quotient is POSITIVE
Different Signs
SUBTRACT and keep the sign of the
LARGER number.
Product or Quotient is NEGATIVE
Do Now:
1) 11 + 15
2) -11 + (-15)
3) 11 + (-15)
4) -11 +15
5) 2 7
6)  2  7
7)  2 7
8) 2  7
9) 8 ∙ ¾
10) – 8 ∙ ¾
11) 8 ∙ - ¾
12) -8 ∙ - ¾
13) 15 ÷ ⅝
14) -15 ÷ ⅝
15) 15 ÷ - ⅝
16) -15 ÷ - ⅝
Subtraction
To subtract signed numbers, change the sign of the number that is being subtracted to its
opposite and then add.
K (keep)
C (change)
O (opposite)
All subtraction problems can be changed to addition problems by changing the “ – “ to “
+ “ and changing the sign of the number that follows the subtraction sign. Then you just
follow the rules of addition.
Example:
Chapter 1, Section 1.4
Page 1
25 – 10 = 15
34 – (-10) = 44
25 – 10 =
K C O
25 + (-10) =
15
34 – (-10) =
K C O
34 + (+10) =
44
Chapter 1: Sets, Operations and Algebraic Language
1.4: Operations with Signed Numbers (Including Absolute Value)
Do Now:
17) 12 - 15
19) -12 – 15
18) 12 - (-15)
Part I:
1) 28 + (-31) =
2) -41.4 + (-19.8) =
3) 23.8 - (-38.3) =
4) -45.07 - (-46.2) =
5) (-7 ½)(5 ⅓) =
6) (-6 ¾)(- 4 ½ ) =
7) (2 ½ ) ÷ (-3 ⅛) =
8) -16 ÷ (- 5 ¼ ) =
20) - 12 – (-15)
Part II:
1) 24 - 12 · 3 + 6 =
a) 6
b) 42
c) -6
d) – 192
c) 183
d) 4,764
2) 36 + 33 ÷ (1/9) - 8 · (12) =
a) 130
b) 171
Absolute value
The absolute value of any number is defined as the number’s distance from zero on the
number line. Since distance is always positive, absolute value is always positive. In
other words, the absolute value of a real number refers to its magnitude or numerical
value without regard to sign.
Imagine a situation: Sarah traveled 14 miles north and 2 miles east. How many miles
total did she travel?
Solution: Since we want only the total miles: 14+2= 16 miles. We ignore the direction.
Absolute value is the same thing, where the “directions” on the number line are indicated
by + or – signs.
Absolute value notation looks like a pair of straightened parentheses.
Chapter 1, Section 1.4
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Chapter 1: Sets, Operations and Algebraic Language
1.4: Operations with Signed Numbers (Including Absolute Value)
Examples:
-7  7
7 7
 7  7
 7  7
When working with absolute value, treat the absolute value sign in the same way that you
would treat parentheses.
 First, simplify the expression inside the absolute value sign.
 Second, make the resulting value positive and drop the absolute value notation.
Absolute Value Notation as a Grouping Symbol
P
E
M
A
D
S
(LEFT TO RIGHT)
The P in PEMDAS stands for parentheses. Parentheses should be interpreted as
including any symbols that group terms together. These include regular or round
parentheses (), box or square parentheses [ ], curly parentheses { } , and angle
parentheses
. Absolute value notation can be thought of as straight parentheses
The fraction bar can also be a grouping symbol (eg.
.
ab
).
cd
Error Alert!
Double negatives and parentheses within parentheses are potential sources of errors when
working with grouping symbols, and especially so when working with absolute value
symbols. Consider the differences between regular parentheses and absolute value
notation in the following examples:
Absolute Value Notation
Curved Parentheses
  3 7 
  3  7  
  3  7 
  21
  21
21
21
Always start with the inner-most parentheses and work your way outward.
Chapter 1, Section 1.4
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Chapter 1: Sets, Operations and Algebraic Language
1.4: Operations with Signed Numbers (Including Absolute Value)
Variables Inside Absolute Value Notation
When a variable is within absolute value notation, and the value of the variable is given,
substitute the given value for the variable and then follow the above steps.
Example:
The expression  7 is equivalent to
(1) 1
(3) 7
(2) 0
(4) –7
One Solution
The expression inside the absolute value sign cannot be simplified, so we make it positive
and drop the absolute value notation. This results in the elimination of everything except
the positive seven and the negative sign in front of it.
 7
  7 
7
Another Regents Problem
If r = 2 and s = -7, what is the value of r  s ?
(1) 5
(3) 9
(2) -5
(4) -9
One Solution
Do the substitution first. Then simplify.
rs
2  7
2   7 
5
Evaluating Algebraic Expressions
An algebraic expression can be evaluated when each of its variables is given a numerical
value.
Evaluate each expression when m = -8 and n = +4.
(a)
mn
mn
Chapter 1, Section 1.4
Page 4
(b)  2n  m
Chapter 1: Sets, Operations and Algebraic Language
1.4: Operations with Signed Numbers (Including Absolute Value)
Solution:
Replace each variable with its assigned value, and then perform the indicated operations.
(a)
mn
mn
8 4
8 4
 12

4
 3

(b)
 2n  m 
 2 4  (8) 
 2 4  (8) 
 2 12
 24
1) Evaluate:
52 ÷ (-22 + 32) + 24 · (1/4) =
2) Evaluate: 122 - 42 ÷ (-1/2) + 2 · (-3)2 =
3) a = -3
b=7
5a - 12b + 9 · 3
2b - 3a + 1
4) Evaluate 3y2 + 8x = , when x = 3 and y= -2
a) 12
b) 36
c) 60
d) 0
5) (112 + 20 ∙ ¾ ) ÷ 4 – 5 ∙ 7 =
6) Evaluate the expression when a = -5, b = 4, and c = -2
3b  16  2c  3a
2a 2  9c  4b


Chapter 1, Section 1.4
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Chapter 1: Sets, Operations and Algebraic Language
1.4: Operations with Signed Numbers (Including Absolute Value)
Chapter 1, Section 1.4
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