1.3 1.7

Adding and Subtracting Real Numbers;
1.3 Properties of Real Numbers

1. Add integers.
2. Add rational numbers.
3. Find the additive inverse of a number.
4. Subtract rational numbers.

Copyright © 2011 Pearson Education, Inc.

Objective 1

Add integers.

Copyright © 2011 Pearson Education, Inc. Slide 1- 2

Parts of an addition statement: The numbers added
are called addends and the answer is called a sum.

2+3=5

Addends Sum


Adding Numbers with the Same Sign

To add two numbers that have the same sign, add
their absolute values and keep the same sign.


Example 2

Add.
a. 27 + 12 b. –16 + (– 22)

Solution
a. 27 + 12 = 39

b. –16 + (–22) = –38


Adding Numbers with Different Signs

To add two numbers that have different signs,
subtract the smaller absolute value from the
greater absolute value and keep the sign of the
number with the greater absolute value.


Example 3

Add.
a. 35 + (–17) b. –29 + 7

Solution
a. 35 + (–17) = 18

b. –29 + 7 = –22


Objective 2

Add rational numbers.


Adding Fractions with the Same Denominator

To add fractions with the same denominator, add
the numerators and keep the same denominator;
then simplify.


Example 4

A
2 4 4  5
+ − +− ÷
9 9 12  12 

4  5
a 2 4 2+4 6 b. − +  − ÷
+ = = 12  12 
9 9 9 9
−4 + ( −5 ) 9
2 g3 2 = =−
= = 12 12
3 g3 3 3 g3 3
Replace 6 and 9 with their prime =− =−
factorizations, divide out the 3 g2 g 2 4
common factor, 3, then multiply Simplify to lowest terms by dividing
the remaining factors. out the common factor, 3.

Adding Fractions

To add fractions with different denominators:
1. Write each fraction as an equivalent fraction
with the LCD.
2. Add the numerators and keep the LCD.
3. Simplify.


Example 5a
1 1
Add: +
3 4
Solution
1 1
+ Write equivalent fractions
with 12 in the denominator.
3 4

Add numerators and keep
the common denominator.

Because the addends have
the same sign, we add and
keep the same sign.


Example 5b
5 3
Add: − +
6 4

Write equivalent fractions
with 12 in the denominator.

Add numerators and keep

Because the addends have
different signs, we subtract and
keep the sign of the number with
the greater absolute value.


Example 5c
7 9
Add: − +
8 30
Solution
7 9 7 ( 15 ) 9(4)
− + =− + Write equivalent fractions
8 30 8 ( 15 ) 30(4) with 120 in the denominator.

105 36
=− + Add numerators and keep
120 120
−105 + 36 Reduce to lowest terms.
=
120
−69 3 ×23 23
= =− =−
120 2 ×2 ×2 × ×
3 5 40


Example 6
Bank account :


Objective 4

Subtract rational numbers.


Parts of a subtraction statement:

8–5=3

Difference
Minuend
Subtrahend


Rewriting Subtraction

To write a subtraction statement as an equivalent
addition statement, change the operation symbol
from a minus sign to a plus sign, and change the
subtrahend to its additive inverse.


Example 9a
Subtract
a. –17 – (–5)

Solution
Write the subtraction as an equivalent addition.
–17 – (–5)
Change the operation Change the subtrahend
from minus to plus. to its additive inverse.
= –17 + 5
= –12


Example 9b
3 1
Subtract: − −
8 4
Solution
3 1 3 1
− − =− −
8 4 8 4
3  1
= − +− ÷
8  4
3  1(2) 
= − +− ÷
Write equivalent fractions with
the common denominator, 8.
8  4(2) 
3  2 5
= − +− ÷ = −
8  8 8

Multiplying and Dividing Real Numbers;
1.4 Properties of Real Numbers

1. Multiply integers.
2. Multiply more than two numbers.
3. Multiply rational numbers.
4. Find the multiplicative inverse of a number.
5. Divide rational numbers.


Objective 1

Multiply integers.


In a multiplication statement, factors are
multiplied to equal a product.

2 g 3 = 6

Factors Product


Multiplying Two Numbers with Different Signs

When multiplying two numbers that have different
signs, the product is negative.


Example 2

Multiply.
a. 7(–4) b. (–15)3

Solution
a. 7(–4) = –28 Warning: Make sure you see the
difference between 7(–4), which
indicates multiplication, and 7 – 4,
b. (–15)3 = –45 which indicates subtraction.


Multiplying Two Numbers with the Same Sign

When multiplying two numbers that have the same
sign, the product is positive.


Example 3

Multiply.
a. –5(–9) b. (–6)(–8)

Solution
a. –5(–9) = 45

b. (–6)(–8) = 48


Objective 2

Multiply more than two numbers.


Multiplying with Negative Factors

The product of an even number of negative factors
is positive, whereas the product of an odd number
of negative factors is negative.


Objective 3

Multiply rational numbers.


Example 5a
3 4
Multiply − g  ÷.
5 9

Solution
3 4 3  2 g2 
− g  ÷= − g  ÷ Divide out the common factor, 3.
5 9 5  3 g3 
4 Because we are multiplying two
=− numbers that have different signs,
15 the product is negative.


Objective 5

Divide rational numbers.


Parts of a division statement:

8 ÷ 2 = 4

Dividend Quotient

Divisor


Dividing Signed Numbers

When dividing two numbers that have the same
sign, the quotient is positive.
When dividing two numbers that have different
signs, the quotient is negative.


Example 8

Divide.
a. 56 ÷ (−8) b. −72 ÷ ( −6 )

Solution
a. 56 ÷ (−8) = −7 b.−72 ÷ ( −6 ) = 12


Division Involving 0

0 ÷ n = 0 when n ≠ 0.
n ÷ 0 is undefined when n ≠ 0.
0 ÷ 0 is indeterminate.


Example 9
3 4
Divide − ÷ .
10 5

Solution
3 4 3 5
− ÷ =− g Write an equivalent multiplication.
10 5 10 4
3 5 Divide out the common factor, 5.
=− g
5 g2 2 g2

3 Because we are dividing two numbers
=− that have different signs, the result is
8 negative.


Exponents, Roots, and Order of
1.5 Operations

1. Evaluate numbers in exponential form.
2. Evaluate square roots.
3. Use the order-of-operations agreement to simplify
numerical expressions.
4. Find the mean of a set of data.


Objective 1

Evaluate numbers in exponential
form.


Evaluating an Exponential Form

To evaluate an exponential form raised to a
natural number exponent, write the base as a
factor the number of times indicated by the
exponent; then multiply.


Example 1a

Evaluate. (–9)2

Solution
The exponent 2 indicates we have two factors of –9.
Because we multiply two negative numbers, the result
is positive.

(–9)2 = (–9)(–9) = 81


Example 1b
3
 3
Evaluate. − ÷
 5

Solution
The exponent 3 means we must multiply the base by
itself three times.
3
 3  3  3   3 
 − ÷ =  − ÷ − ÷ − ÷
 5  5  5   5 

27
=−
125


Evaluating Exponential Forms with Negative
Bases

If the base of an exponential form is a negative
number and the exponent is even, then the
product is positive.
If the base is a negative number and the exponent is
odd, then the product is negative.


Example 2
Evaluate.
a. (−3) 4 b. −34 c. (−2)3 d. −23


Objective 2

Evaluate square roots.


Roots are inverses of exponents. More
specifically, a square root is the inverse of a
square, so a square root of a given number is a
number that, when squared, equals the given
number.

Square Roots
Every positive number has two square roots, a
positive root and a negative root.
Negative numbers have no real-number square
roots.


Example 3

Find all square roots of the given number.

Solution
a. 49
Answer ± 7

b. −81
Answer No real-number square roots exist.


The symbol, , called the radical, is used to
indicate finding only the positive (or principal)
square root of a given number. The given number or
expression inside the radical is called the radicand.
Radical
Principal Square Root

25 = 5

Radicand


Example 4

Evaluate the square root.
a. 169 b. 64 c. 0.64 d. −25
81

Solution


Objective 3

Use the order-of-operations
agreement to simplify numerical
expressions.


Order-of- Operations Agreement

Perform operations in the following order:
1. Within grouping symbols: parentheses ( ),
brackets [ ], braces { }, above/below fraction
bars, absolute value | |, and radicals .
2. Exponents/Roots from left to right, in order as
they occur.
3. Multiplication/Division from left to right, in order
as they occur.
4. Addition/Subtraction from left to right, in order as
they occur.

Example 5a

Simplify. −26 + 15 ÷ (−5) ×2

Solution
−26 + 15 ÷ (−5) ×2

= −26 + (−3) ×2 Divide 15 ÷ (−5) = –3

= −26 + (−6) Multiply (–3) ⋅ 2 = –6

= −32 Add –26 + (–6) = –32


Example 5c

Simplify. ( −3) + 5 6 − ( 2 + 1)  − 49
2
 

Solution
Calculate within the innermost
parenthesis.

Evaluate the exponential form,
brackets, and square root.

Multiply 5(3).

Add 9 + 15.

Subtract 24 – 7.


Example 7a
8(−5) − 23
Simplify. 4(8) − 8

Solution
Evaluate the exponential form in
the numerator and multiply in the
denominator.

Multiply in the numerator and
subtract in the denominator.

Subtract in the numerator.

Divide.

Example 7b
9(4) + 12
Simplify. 43 + (8)(−8)

Solution

Because the denominator or divisor is 0, the answer is
undefined.


1.6 Translating Word Phrases to Expressions

1. Translate word phrases to expressions.


Objective 1

Translating word phrases to
Expressions

Look at the pg


The key words sum, difference, product, and quotient
indicate the answer for their respective operations.

sum of x and 3 difference of x and 3

x+3 x–3

product of x and 3 quotient of x and 3

x⋅3 x÷3


Example 1

Translate to an algebraic expression.
a. five more than two times a number
Translation: 5 + 2n or 2n + 5

b. seven less than the cube of a number
Translation: n3 – 7

c. the sum of h raised to the fourth power and twelve
Translation: h4 + 12


Example 2

Translate to an algebraic expression.
a. seven times the sum of a and b
Translation: 7(a + b)

b. the product of a and b divided by the sum
of w2 and 4
ab
Translation: ab ÷ (w + 4) or
2
2
w +4


1.7 Evaluating and Rewriting Expressions

1. Evaluate an expression.
2. Determine all values that cause an expression to be
undefined.
3. Rewrite an expression using the distributive property.
4. Rewrite an expression by combining like terms.


Objective 1

Evaluate an expression.


Evaluating an Algebraic Expression

To evaluate an algebraic expression:
1. Replace the variables with their corresponding
given values.
2. Calculate the numerical expression using the order
of operations.


Example 1a
Evaluate 3w – 4(a – 6) when w = 5 and a = 7.

Solution
3w – 4(a − 6)

3(5) – 4(7 – 6) Replace w with 5 and a with 7.

= 3(5) – 4(1) Simplify inside the parentheses first.

= 15 – 4 Multiply.

= 11 Subtract.


Objective 2

Determine all values that cause an
expression to be undefined.


Examples:


The Distributive Property of Multiplication over
Addition
a(b + c) = ab + ac

This property gives us an alternative to the order of
operations.
2(5 + 6) = 2(11) 2(5 + 6) = 2⋅5 + 2⋅6
= 22 = 10 + 12
= 22


1.3 1.7

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