Download Operations with Integers/Add and Subtract Rational Numbers

OPERATIONS WITH
INTEGERS, ADDING AND
SUBTRACTING RATIONAL
NUMBERS
Objective: To add, subtract, multiply, and divide integers, to compare and order rational
numbers, and to add and subtract rational numbers.
ADDING INTEGERS
 The sum of two positive numbers is a positive number.
 The sum of two negative numbers is a negative number.
 The sum of a positive number and a negative number will either
be positive or negative depending on the sign of the number with the
larger absolute value.
ADDING INTEGERS
 4 + 5 = 9 (Positive + Positive = Positive)
 -4 + -5 = -9 (Negative + Negative = Negative)
 -4 + 5 = 1 (Negative + Positive = Positive because the Positive is
the larger absolute value)
 4 + -5 = -1 (Positive + Negative = Negative because the Negative
is the larger absolute value)
ADDING INTEGERS
 You may add integers using a number line.
 Always start at zero.
 Positive numbers move right and negative numbers move left.
EXAMPLE 1
 Use a number line to find -2 + 3. = 1
-2



3




ADDING INTEGERS
 You may also add integers by using absolute values.
 If the two numbers have the same sign (both positive or both
negative), add the absolute value of the two numbers and match the
sign to the two numbers.
 If the two numbers have different signs (one positive and one
negative), subtract the absolute value of the two numbers and match
the sign of the larger absolute value number.
EXAMPLE 2
 Add the following.
a. -15 + (-4) = -19
b. -15 + 4 = -11
c. 15 + -4 = 11
SUBTRACTING INTEGERS
 Every positive integer can be paired with a negative integer. These
pairs are called opposites.
 A number and its opposite are additive inverses. Additive inverses
can be used when you subtract integers.
 Subtracting integers is the same as adding the inverse.
 a – b = a + (-b)
EXAMPLE 3
 Subtract the following.
a. 11 – 15 = 11 + (-15) = -4
b. -11 – 15 = -11 + (-15) = -26
c. 11 – (-15) = 11 + 15 = 26
M U LT I P LY IN G A N D D I V I D I NG
INTEGERS
 If you multiply or divide integers with the same sign, your answer
will be positive.
•
•
•
•
Positive x Positive = Positive
Negative x Negative = Positive
Positive ÷ Positive = Positive
Negative ÷ Negative = Positive
M U LT I P LY IN G A N D D I V I D I NG
INTEGERS
 If you multiply or divide integers with different signs, your answer
will be negative.
•
•
•
•
Positive x Negative = Negative
Negative x Positive = Negative
Positive ÷ Negative = Negative
Negative ÷ Positive = Negative
EXAMPLE 4
 Find each product or quotient.
a. -9(4) = -36
b. -112 ÷ (-8) = 14
c. -11(-11) = 121
d. 96 ÷ (-6) = -16
COMPARING RATIONAL
NUMBERS
 You can compare rational numbers in two different ways.
 When comparing fractions, always make them have common
denominators or turn them into decimals and compare the decimals.
EXAMPLE 5
4
5
 Replace  with >, <, or = to make  a true statement.
5
8
>
4 32

5
40
5 25

8
40
EXAMPLE 6
3
3
2
 Order 3 ,  3 , 3.45, and  3 from least to greatest.
8
7
5
3
3  3 15
7
35
2
3  3 14
5
35
100N = 45.4545. . .
-N = 0.4545. . .
99N = 45
45 5
N 
99 11
3
2 3
3 ,  3 , 3 , 3.45
7
5 8
5
 340
11
88
3
3  3 33
8
88
3
A D D I N G O R S U B T R AC T I N G F R AC T I O N S
WITH LIKE DENOMINATORS
 To add or subtract fractions with the same denominator, add or
subtract the numerators and write the sum or difference over the
common denominator.
 Follow the addition and subtraction rules for integers to determine
the sign of your answer.
EXAMPLE 7
 Find each sum or difference. Write in simplest form.
5 3
8
a. . 

11 11 11
b. 11
.  7  4 2
18 18 18 9
3 5
2
1
c. .     
8 8
8
4
A D D I N G O R S U B T R AC T I N G F R AC T I O N S
WITH UNLIKE DENOMINATORS
 To add or subtract fractions with unlike denominators, first find
the least common denominator (LCD).
 Rename each fraction with the LCD, and then add or subtract.
 Simplify if possible.
EXAMPLE 8
 Find each sum or difference. Write in simplest form.
4 7
15 3
a. . 
 8  7 

5 10 10 10 10 2
5 5 15 5 10 5
  

6 18 18 18 18 9
b. . 
c.
3 3 6 15
9
   
10 4 20 20
20
.
EXAMPLE 9
 Use a number line to find -1.25 + 1.5. = 0.25
1.5
-1.25





EXAMPLE 10
 Find each sum.
a.
-12.6 + (-3.9) = -16.5
b. 
. 2  5  6  5  1
3
9
9
9
9
EXAMPLE 11
 Find -17.55 – (-24.5).
= -17.55 + 24.5
= 6.95