1 Evaluate absolute values.
2 Add real numbers.
3 Subtract real numbers.
4 Multiply real numbers.
5 Divide real numbers.
6 Use the properties of real numbers.
Two numbers that are the same distance from 0 on the number line but in opposite directions are called additive inverses, or opposites, of each other. For example, 3 is the additive inverse of -3, and -3 is the additive inverse of 3. The number 0 is its own ad- ditive inverse. The sum of a number and its additive inverse is 0. What are the additive inverses of -56.3 and 76
5 ? Their additive inverses are 56.3 and -76
5, respectively.
Additive Inverses -56.3 and 56.3
76
5 and -76 5 0 and 0
Notice that the additive inverse of a positive number is a negative number and the additive inverse of a negative number is a positive number.
Additive Inverse
For any real number a, its additive inverse is -a.
Consider the number -5. Its additive inverse is -1-52. Since we know this number must be positive, this implies that -1-52 = 5. This is an example of the double negative property.
Double Negative Property
For any real number a, -1-a2 = a.
By the double negative property, -1-7.42 = 7.4 and -a-12 5 b = 12
5. Understanding Algebra
The additive inverse
• of a positive number is a negative number.
• of a negative number is a positive number.
• of zero is zero.
1 Evaluate Absolute Values
Absolute Value
The absolute value of a number is its distance from the number 0 on the real number line.
The symbol 0 0 is used to indicate absolute value.
Consider the numbers 3 and -3 (Fig. 1.6). Both numbers are 3 units from 0 on the number line. Thus
030 = 3 and 0-30 = 3
EXAMPLE 1 Evaluate.
a) 070 b) 0-110 c) 000 d) `-2
3` e) 06.50 Solution
a) 070 = 7, since 7 is 7 units from 0 on the number line.
b) 0-110 = 11, since -11 is 11 units from 0 on the number line.
c) 000 = 0, since 0 is 0 units from 0 on the number line.
d) `-2 3` = 2
3, since -2 3 is 2
3 of a unit from 0 on the number line.
e) 06.50 = 6.5, since 6.5 is 6.5 units from 0 on the number line.
Now Try Exercise 13 To determine the absolute value of a real number without using a number line, use the following definition.
Understanding Algebra
The absolute value of any nonzero number will always be a positive number, and the absolute value of 0 is 0.
Absolute Value
If a represents any real number, then
0a0 = b a if a Ú 0
-a if a 6 0
The absolute value of any nonnegative number is the number itself. The absolute value of a negative number is the additive inverse, or opposite, of the number. We use the definition of absolute value to evaluate several expressions in our next example.
EXAMPLE 2 Evaluate using the definition of absolute value.
a) 000 b) 030 c) 0-50 d) -030 e) -0-50 Solution
a) Since 0 is nonnegative, 000 = 0.
b) Since 3 is nonnegative, 030 = 3.
c) Since -5 is negative, 0-50 = -1-52 = 5.
d) We are asked to determine the additive inverse of 030. In part b) we determined that 030 = 3. Thus, -030 = -3.
5 6
4 3 2 1 0 24
25
26 23 22 21
3 units 3 units
FIGURE 1.6
2 Add Real Numbers
To Add Two Numbers with the Same Sign (Both Positive or Both Negative) Add their absolute values and place the common sign before the sum.
The sum of two positive numbers will always be a positive number, and the sum of two nega- tive numbers will always be a negative number.
EXAMPLE 4 Evaluate -4 + 1-72.
Solution Since both numbers being added are negative, the sum will be negative.
We need to add the absolute values of these numbers and then place a negative sign before the value. First, determine the absolute value of each number.
0-40 = 4 0-70 = 7
Then add the absolute values.
0-40 + 0-70 = 4 + 7 = 11
Finally, since both numbers are negative, the sum must be negative. Thus, -4 + 1-72 = -11
Now Try Exercise 45
To Add Two Numbers with Different Signs (One Positive and the Other Negative)
Subtract the smaller absolute value from the larger absolute value. The answer has the sign of the number with the larger absolute value.
The sum of a positive number and a negative number may be either positive, negative, or zero.
The sign of the answer will be the same as the sign of the number with the larger absolute value.
EXAMPLE 5 Evaluate 5 + 1-92.
Solution Since the numbers being added are of opposite signs, we subtract the smaller absolute value from the larger absolute value. First we take each absolute value.
050 = 5 0-90 = 9
Now we determine the difference, 9 - 5 = 4. The number -9 has a larger absolute value than the number 5, so their sum is negative.
5 + 1-92 = -4
Now Try Exercise 43 e) We are asked the determine the additive inverse of 0-50. In part c) we determined
that 0-50 = 5. Thus, -0-50 = -5.
Now Try Exercise 19
Understanding Algebra
• The sum of two positive numbers will always be a positive number.
• The sum of two negative numbers will always be a negative number.
• The sum of a positive num- ber and a negative number may be either positive, negative, or zero.
EXAMPLE 3 Insert 6, 7, or = in the shaded area between the two values to make each statement true.
a) 080 0-80 b) 0-10 -0-30 Solution
a) Since both 080 and 0-80 equal 8, we have 080 = 0-80. b) Since 0-10 = 1 and -0-30 = -3, we have 0-10 7 -0-30.
Now Try Exercise 29
In part b) of our next example, we will be adding fractions with different denomina- tors. To do this we will need to rewrite the fractions with the least common denominator.
Least Common Denominator
The least common denominator (LCD) of a set of denominators is the smallest number that each denominator divides into without remainder.
In Example 6 b), below, the two denominators are 8 and 6. The LCD of these denomi- nators is 24, since 24 is the smallest number that both 8 and 6 divide without remainder.
EXAMPLE 6 Evaluate. a) 1.3 + 1-2.72 b) -7 8 + 5
6
Solution
a) 1.3 + 1-2.72 = -1.4
b) Begin by writing both fractions with the least common denominator, 24.
-7 8 + 5
6 = -21 24 + 20
24 = 1-212 + 20
24 = -1
24 = - 1 24
Now Try Exercise 51
EXAMPLE 7 Depth of Ocean Trenches The Palau Trench in the Pacific Ocean lies 26,424 feet below sea level. The deepest ocean trench, the Mariana Trench, is 9416 feet deeper than the Palau Trench (see Fig. 1.7). Determine the depth of the Mariana Trench.
Palau trench 26,424 ft Depth Below Sea Level
Mariana trench 9416 ft deeper
Feet (thousands)
215 210 25
220 225 230 235 240 245
FIGURE 1.7
Solution Consider distance below sea level to be negative. Therefore, the total depth is -26,424 + 1-94162 = -35,840 feet
or 35,840 feet below sea level.
Now Try Exercise 127
3 Subtract Real Numbers
Every subtraction problem can be expressed as an addition problem using the following rule.
Subtraction of Real Numbers
a - b= a + 1-b2
To subtract b from a, add the opposite (or additive inverse) of b to a.
For example, 5 - 7 means 5 - 1+72. To subtract 5 - 7, add the opposite of +7, which is -7, to 5.
5 - 7 = 5 + 1-72 c c c c subtract positive add negative
7 7 Since 5 + 1-72 = -2, then 5 - 7 = -2.
EXAMPLE 8 Evaluate.
a) 8 - 3 b) 2 - 9 c) -5 - 2 d) -5 - 8
Solution
a) 8 - 3 = 8 + 1-32 = 5 b) 2 - 9 = 2 + 1-92 = -7 c) -5 - 2 = -5 + 1-22 = -7 d) -5 - 8 = -5 + 1-82 = -13
Now Try Exercise 55
EXAMPLE 9 Evaluate 8 - 1-152.
Solution In this problem, we are subtracting a negative number. The procedure to subtract remains the same.
8 - 1-152 = 8 + 15 = 23 c c c c subtract negative add positive
15 15 Thus, 8 - 1-152 = 23.
Now Try Exercise 47
From Example 9 and other examples, we have the following principle.
Subtracting a Negative Number
a- 1-b2 = a + b
We can use this principle to evaluate problems such as 8 - 1-152 and other problems where we subtract a negative quantity.
EXAMPLE 10 Evaluate -17 - 1-82.
Solution - 17 - 1 - 8 2 = - 17 + 8 = - 9
Now Try Exercise 49
EXAMPLE 11 a) Subtract 35 from -42. b) Subtract -3
5 from -5 9.
Solution
a) -42 - 35 = -77 b) -5
9 - a- 3
5b = -5 9 + 3
5 = -25 45 + 27
45 = 2 45
Now Try Exercise 99
EXAMPLE 12 Extreme Temperatures The hottest temperature ever recorded in the United States was 134°F, which occurred at Greenland Ranch, California, in Death Valley on July 10, 1913. The coldest temperature ever recorded in the United States was -79.8°F, which occurred at Prospect Creek Camp, Alaska, in the Endicott Mountains on January 23, 1971 (see Fig. 1.8). Determine the difference between these two temperatures.
1348 CA
AK 279.88
Degrees Fahrenheit
290 260 230 30 60 90 120 150
FIGURE 1.8
Solution To determine the difference, we subtract.
134° - 1-79.8°2 = 134° + 79.8° = 213.8°
Now Try Exercise 125 Addition and subtraction are often combined in the same problem, as in the follow- ing examples. Unless parentheses are present, if the expression involves only addition and subtraction, we add and subtract from left to right. When parentheses are used, we add and subtract within the parentheses first. Then we add and subtract from left to right.
EXAMPLE 13 Evaluate -15 + 1-372 - 15 - 92.
Solution -15 + 1-372 - 15 - 92 = -15 + 1-372 - 1-42
= -15 - 37 + 4
= -52 + 4 = -48
Now Try Exercise 87
EXAMPLE 14 Evaluate 2 - 0-30 + 4 - 16 - 0-802.
Solution Begin by replacing the numbers in absolute value signs with their numerical equivalents; then evaluate.
2 - 0-30 + 4 - 16 - 0-802 = 2 - 3 + 4 - 16 - 82
= 2 - 3 + 4 - 1-22
= 2 - 3 + 4 + 2
= -1 + 4 + 2
= 3 + 2 = 5
Now Try Exercise 59
4 Multiply Real Numbers
Multiply Two Real Numbers
1. To multiply two numbers with like signs, either both positive or both negative, multiply their absolute values. The product is positive.
2. To multiply two numbers with unlike signs, one positive and the other negative, multiply their absolute values. The product is negative.
Understanding Algebra
The product of two numbers with
• like signs is positive.
• unlike signs is negative.
When multiplying more than two numbers, the product will be negative when there is an odd number of negative numbers. The product will be positive when there is an even number of negative numbers.
EXAMPLE 15 Evaluate. a) 14.221-1.62 b) 1-182 a-1 2b.
Solution
a) 14.221-1.62= -6.72 The numbers have unlike signs. The product is negative.
b) 1-182 a-1
2b = 9 The numbers have like signs, both negative. The product is positive.
Now Try Exercise 65
EXAMPLE 16 Evaluate 41-221-32112.
Solution 41-221-32112 = 1-821-32112 = 24112 = 24
Now Try Exercise 67
Understanding Algebra
When multiplying more than two negative numbers, the product will be
• negative, if there are an odd number of negative numbers.
• positive, if there are an even number of negative numbers.
Multiplicative Property of Zero For any number a,
a#0 = 0#a = 0
By the multiplicative property of zero, 5102 = 0 and 1-7.32102 = 0.
EXAMPLE 17 Evaluate -6#7#0#1-2.972# a-3 5b.
Solution If one or more of the factors is 0, the entire product equals 0. Thus, -6#7#0#1-2.972#a-3
5b = 0.
Now Try Exercise 101
5 Divide Real Numbers
The rules for the division of real numbers are similar to those for multiplication of real numbers.
Understanding Algebra
The quotient of two numbers with
• like signs is positive.
• unlike signs is negative.
Divide Two Real Numbers
1. To divide two numbers with like signs, either both positive or both negative, divide their absolute values. The quotient is positive.
2. To divide two numbers with unlike signs, one positive and the other negative, divide their absolute values. The quotient is negative.
EXAMPLE 18 Evaluate. a) -24 , 4 b) -6.45 , 1-0.42
Solution
a) -24
4 = -6 The numbers have unlike signs. The quotient is negative.
b) -6.45
-0.4 = 16.125 The numbers have like signs. The quotient is positive.
Now Try Exercise 71
EXAMPLE 19 Evaluate a-3
8 b , `-2 5 `.
Solution Since `-2
5 ` is equal to 2
5, we write -3
8 , `-2 5 ` = -3
8 , 2 5 Now multiply by the reciprocal of the divisor.
-3 8 , 2
5 = -3
8 #52 = -3#5
8#2 = -1615 or -1516
Now Try Exercise 75 When the denominator of a fraction is a negative number, we usually rewrite the fraction with a positive denominator. To do this, we use the following fact.
Sign of a Fraction
For any number a and any nonzero number b, a -b = -a
b = -a b Understanding Algebra
A negative fraction can have the minus sign in the denominator, or in the numerator, or in front of the fraction. Thus,
3 -4 = -3
4 = - 3 4.
Thus, when we have a quotient of 1
-2, we rewrite it as either -1 2 or -1
2.
6 Use the Properties of Real Numbers
We have already discussed the double negative property and the multiplicative property of zero. Table 1.1 lists other basic properties for the operations of addition and multiplica- tion on the real numbers.
Table 1.1
FOR REAL NUMBERS a, b, and c ADDITION MULTIPLICATION
Commutative property a + b = b + a ab = ba
Associative property 1a + b2 + c = a + 1b + c2 1ab2c = a1bc2
Identity property a + 0 = 0 + a = a
¢0 is called the additive identity element. ≤
a#1 = 1#a = a
¢1 is called the multiplicative identity element. ≤
Inverse property a + 1-a2 = 1-a2 + a = 0
¢-a is called the additive inverse or opposite of a.≤
a#1
a = 1 a#a = 1
° 1
a is called the multiplicative inverse or reciprocal of a, a ≠ 0.
≥ Distributive property (of
multiplication over addition) a1b + c2 = ab + ac
Note that the commutative property involves a change in order, and the associative property involves a change in grouping.
The distributive property also applies when there are more than two numbers within the parentheses.
a1b + c + d + g+ n2 = ab + ac + ad + g+ an
EXAMPLE 20 Name each property illustrated.
a) x#2 = 2#x b) 1x + 52 + 7 = x + 15 + 72
c) 6 + x = x + 6 d) 214x2 = 12#42x e) 21x + y2 = 2x + 2y
Solution
a) The order of multiplication changed. Thus, x#2 = 2#x demonstrates the commuta- tive property of multiplication.
b) The grouping of addition changed. Thus, 1x + 52 + 7 = x + 15 + 72 demon- strates the associative property of addition.
c) The order of addition changed. Thus, 6 + x = x + 6 demonstrates the commutative property of addition.
d) The grouping of multiplication changed. Thus, 214x2 = 12#42x demonstrates the associative property of multiplication.
e) The number 2 is distributed to x and to y. Thus, 21x + y2 = 2x + 2y demonstrates the distributive property.
Now Try Exercise 113
EXAMPLE 21 Name each property illustrated.
a) p#1 = p b) 1
2 + 0 = 1
2 c) 6 + 1-62 = 0 d) 7#17 = 1
Solution
a) Identity property of multiplication b) Identity property of addition c) Inverse property of addition d) Inverse property of multiplication
Now Try Exercise 115