Transcript 1.2 Operations with Real Numbers and Simplifying Algebraic

§ 1.2
Operations with Real Numbers and
Simplifying Algebraic Expressions
Finding Absolute Value
Absolute value is used to describe how to operate with positive and
negative numbers.
Geometric Meaning of Absolute Value
The absolute value of a real number a, denoted a ,
is the distance from 0 to a on the number line.
This distance is always nonnegative.
 5  5
3  3
The absolute value of -5 is 5 because -5 is 5 units
from 0 on the number line.
The absolute value of 3 is +3 because 3 is 3 units
from 0 on the number line.
Blitzer, Algebra for College Students, 6e – Slide #2 Section 1.2
Rules for Addition of Real Numbers
To add two real numbers with like signs, add their absolute
values. Use the common sign as the sign of the sum.
To add two real numbers with different signs, subtract the
smaller absolute value from the greater absolute value. Use
the sign of the number with the greater absolute value as the
sign of the sum.
Blitzer, Algebra for College Students, 6e – Slide #3 Section 1.2
Adding Real Numbers
EXAMPLE
Add: -12+(-5)
Answer: -17
We are adding numbers having like signs. So
we just add the absolute values and take the
common sign as the sign of the sum.
EXAMPLE
We are adding numbers having unlike signs.
Add: -10 +14 We just take the difference of the absolute
values (difference is 4) and then take the sign
Answer: +4 of the number that has the largest absolute
value (that’s the 14 and it is positive).
Blitzer, Algebra for College Students, 6e – Slide #4 Section 1.2
Adding Real Numbers
EXAMPLE
2
5
Add:  
3
20
SOLUTION
2 3
 
5 20
 2
3 
   

 5 20 
2 3 
  
 5 20 
The two numbers in this example have
different signs. We know that 2/5 > 3/20. We
need to subtract the smaller absolute value
from the larger and take the sign of the
number having the greater absolute value.
Our answer will be negative since the sign of
2/5 is negative.
Using the rule, rewrite with absolute
values.
Then simplify.
Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.2
Adding Real Numbers
CONTINUED
2 4 3 
   
 5 4 20 
Common denominators
3 
 8
  
 20 20 
Multiply
 5 
 
 20 
Subtract
1

4
Finally, simplify the fraction.
Whew! This last example was a little difficult. In practice, we
don’t always rewrite using the absolute values. We just learn the
rules and carry out the computation without putting in all the
formal steps.
Blitzer, Algebra for College Students, 6e – Slide #6 Section 1.2
Subtracting Real Numbers
Definition of Subtraction
If a and b are real numbers,
a – b = a + (-b)
That is, to subtract a number, just add its additive
opposite (called its additive inverse).
Blitzer, Algebra for College Students, 6e – Slide #7 Section 1.2
Subtracting Real Numbers
EXAMPLE
Subtract: -12-(-5)
-12-(-5)
-12+5
-7
Here, change the subtraction to
addition and replace -5 with its
additive opposite. That is, replace
the -(-5) with 5.
EXAMPLE
Subtract: -10 - (+4)
-10 +(-4)
-14
Here, change the subtraction to addition
and replace +4 with its additive opposite
of -4. Then you use the rule for adding
two negative numbers.
Blitzer, Algebra for College Students, 6e – Slide #8 Section 1.2
Multiplying Real Numbers
Rule
The product of two real numbers with different
signs is found by multiplying their absolute
values. The product is negative.
The product of two real numbers with the same
sign is found by multiplying their absolute
values. The product is positive.
The product of 0 and any real number is 0
If no number is 0, a product with an odd
number of negative factors is found by
multiplying absolute values. The product is
negative.
If no number is 0, a product with an even
number of negative factors is found by
multiplying absolute values. The product is
positive.
Examples
(-4)8 = -32
(-2)(-11) = -22
0(-14) = 0
(-3)(-10)(-6) = -180
-4(-8)5 = 160
Blitzer, Algebra for College Students, 6e – Slide #9 Section 1.2
Dividing Real Numbers
Rules for Dividing Real Numbers
The quotient of two numbers with different signs is
negative.
The quotient of two numbers with the same sign is
positive.
In either multiplication or division of signed numbers, it is important
to count the negatives in the product or quotient:
Odd number of negatives and the answer is negative.
Even number of negatives and the answer is positive.
Blitzer, Algebra for College Students, 6e – Slide #10 Section 1.2
Dividing Real Numbers
EXAMPLE
Divide.
5
1

3
4
SOLUTION
5 4
  
3 1
5
1

3
4
5
1
   
4
3
5 1
  
3 4



 5 4 


 3 1 
20

3
Blitzer, Algebra for College Students, 6e – Slide #11 Section 1.2
Order of Operations
EXAMPLE
Simplify.
4  32
6
2
6
SOLUTION
4  32
6
2
6
6
49
2
6
6
36
2
6
662
2
Evaluating exponent
Multiply
Divide
Subtract
Blitzer, Algebra for College Students, 6e – Slide #12 Section 1.2
Basic Algebraic Properties
Property
Commutative
Examples
2+3=3+2
10 + 4 = 4 + 10
8+7=7+8
2(3) = 3(2)
4(10) = 10(4)
7(8) = 8(7)
Associative
4 + (3 + 2) = (4 + 3) + 2
(6  4)11 = 6(4  11)
3(2  5) = (3  2)5
Distributive
7(2x + 3) = 14x + 21
5(3x-2-4y) = 15x – 10 – 20y
(2x + 7)4 = 8x + 28
Blitzer, Algebra for College Students, 6e – Slide #13 Section 1.2
Combining Like Terms
EXAMPLE
Simplify: 3a – (2a + 4b – 6c) +2b – 3c
SOLUTION
3a – (2a + 4b – 6c) +2b – 3c
3a – 2a - 4b + 6c +2b – 3c
Distributive Property
(3a – 2a) + (2b - 4b) + (6c – 3c)
Comm. & Assoc. Prop.
(3 – 2)a + (2 - 4)b + (6 – 3)c
Distributive Property
1a - 2b + 3c
Subtract
Simplify
a - 2b + 3c
Blitzer, Algebra for College Students, 6e – Slide #14 Section 1.2