§ 1.2 Operations with Real Numbers and Simplifying Algebraic Expressions.
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Transcript of § 1.2 Operations with Real Numbers and Simplifying Algebraic Expressions.
Blitzer, Algebra for College Students, 6e – Slide #2 Section 1.2
Finding Absolute Value
Absolute value is used to describe how to operate with Absolute value is used to describe how to operate with positive and negative numbers. positive and negative numbers.
55
33
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.
Geometric Meaning of Absolute ValueThe 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.
Blitzer, Algebra for College Students, 6e – Slide #3 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 #4 Section 1.2
Adding Real Numbers
Add: -12+(-5)
EXAMPLEEXAMPLE
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.
Answer: -17
EXAMPLEXAMPLEE
Add: -10 +14We are adding numbers having unlike signs. We just take the difference of the absolute values (difference is 4) and then take the sign of the number that has the largest absolute value (that’s the 14 and it is positive).
Answer: +4
Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.2
Adding Real Numbers
Add:
EXAMPLEEXAMPLE
20
3
5
2
SOLUTIONSOLUTION
20
3
5
2
20
3
5
2
20
3
5
2
Using the rule, rewrite with absolute values.
Then simplify.
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.
Blitzer, Algebra for College Students, 6e – Slide #6 Section 1.2
Adding Real Numbers
Common denominators
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.
CONTINUECONTINUEDD
20
3
4
4
5
2
20
3
20
8
20
5
4
1
Multiply
Subtract
Blitzer, Algebra for College Students, 6e – Slide #7 Section 1.2
Subtracting Real Numbers
Definition of SubtractionDefinition 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 #8 Section 1.2
Subtracting Real Numbers
Subtract: -12-(-5)
EXAMPLEEXAMPLE
-12+5
-7
Here, change the subtraction to addition and replace -5 with its
additive opposite. That is, replace the -(-5) with 5.
-12-(-5)
EXAMPLEEXAMPLE
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 #9 Section 1.2
Multiplying Real Numbers
Rule ExamplesThe product of two real numbers with different signs is found by multiplying their absolute values. The product is negative.
(-4)8 = -32
The product of two real numbers with the same sign is found by multiplying their absolute values. The product is positive.
(-2)(-11) = -22
The product of 0 and any real number is 0 0(-14) = 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.
(-3)(-10)(-6) = -180
If no number is 0, a product with an even number of negative factors is found by multiplying absolute values. The product is positive.
-4(-8)5 = 160
Blitzer, Algebra for College Students, 6e – Slide #10 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 importantto 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 #11 Section 1.2
Dividing Real Numbers
EXAMPLEEXAMPLE
4
1
3
5Divide.
4
1
3
5
4
1
3
5
4
1
3
5
1
4
3
5
13
45
3
20
SOLUTIONSOLUTION
Blitzer, Algebra for College Students, 6e – Slide #12 Section 1.2
Order of Operations
EXAMPLEEXAMPLE
Simplify. 26
346
2
26
346
2
SOLUTIONSOLUTION
26
946
26
366
266
2
Evaluating exponent
Multiply
Divide
Subtract
Blitzer, Algebra for College Students, 6e – Slide #13 Section 1.2
Basic Algebraic Properties
Property Examples
Commutative
2 + 3 = 3 + 2 2(3) = 3(2)
10 + 4 = 4 + 10 4(10) = 10(4)
8 + 7 = 7 + 8 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 #14 Section 1.2
Combining Like Terms
EXAMPLEEXAMPLE
Simplify: 3a – (2a + 4b – 6c) +2b – 3c
3a – (2a + 4b – 6c) +2b – 3c
SOLUTIONSOLUTION
3a – 2a - 4b + 6c +2b – 3c
(3a – 2a) + (2b - 4b) + (6c – 3c)
(3 – 2)a + (2 - 4)b + (6 – 3)c
1a - 2b + 3c
Distributive Property
Comm. & Assoc. Prop.
Distributive Property
a - 2b + 3c
Subtract
Simplify