§ 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