Chapter 1 Section 5

Adding and Subtracting Real Numbers 1.5 Adding and Subtracting Real Numbers Add two numbers with the same sign. Add two numbers with different signs. Use the definition of subtraction. Use the rules for order of operations with real numbers. Translate words and phrases involving addition and subtraction. Use signed numbers to interpret data. 2 3 4 5 6

Add two numbers with the same sign. Objective 1 Add two numbers with the same sign. Slide 1.5-3

Add two numbers with the same sign. The sum of two negative numbers is a negative number whose distance from 0 is the sum of the distance of each number from 0. That is, the sum of two negative numbers is the negative of the sum of their absolute values. Adding Numbers with the Same Sign To add two numbers with the same sign, add the absolute values of the numbers. The sum has the same sign as the numbers being added. Example: To avoid confusion, two operation symbols should not be written successively without a parenthesis between them. Slide 1.5-4

5 −7 EXAMPLE 1 Adding Numbers on a Number Line Use a number line to find each sum. Solution: 5 −7 Slide 1.5-5

Adding Two Negative Numbers EXAMPLE 2 Adding Two Negative Numbers Find the sum. Solution: Slide 1.5-6

Add two numbers with different signs. Objective 2 Add two numbers with different signs. Slide 1.5-7

Adding Numbers with the Same Sign Add two numbers with different signs. Adding Numbers with the Same Sign To add two numbers with different signs, find the absolute values of the numbers and subtract the lesser absolute value from the greater. Give the answer the same sign as the number having the greater absolute value. For instance, to add −12 and 6, find their absolute values: and Then find the difference between these absolute values: The sum will be negative, since , so the final answer is . Slide 1.5-8

3 EXAMPLE 3 Adding Numbers with Different Signs Use a number line to find the sum. Solution: 3 Slide 1.5-9

Adding Numbers with Different Signs EXAMPLE 4 Adding Numbers with Different Signs Find the sum. Solution: Find their absolute values: and Then find the difference between these absolute values: The sum will be negative, since , so the final answer is . Slide 1.5-10

EXAMPLE 5 Adding Mentally Check each answer. Solution: Correct Correct Slide 1.5-11

Use the definition of subtraction. Objective 3 Use the definition of subtraction. Slide 1.5-12

Use the definition of subtraction. The answer to a subtraction problem is called a difference. In the subtraction x −y, x is called the minuend and y is called the subtrahend. We can illustrate the subtraction of 4 from 7, written 7 − 4, with a number line. The procedure to find the difference 7 − 4 is exactly the same procedure that would be used to find the sum. Slide 1.5-13

Use the definition of subtraction. (cont’d) The previous equation suggests that subtracting a positive number from a greater positive number is the same as adding the additive inverse of the lesser number to the greater. Definition of Subtraction For any real numbers x and y, To subtract y from x, add the additive inverse (or opposite) of y to x. That is, change the subtrahend to its opposite and add. Slide 1.5-14

Using the Definition of Subtraction EXAMPLE 6 Using the Definition of Subtraction Subtract. Solution: Slide 1.5-15

Use the definition of subtraction. (cont’d) Uses of the Symbol − We use the symbol − for three purposes: 1. to represent subtraction, as in 2. to represent negative numbers, such as −10, −2, and −3; 3. to represent the opposite (or negative) of a number, as in “the opposite (or negative) of 8 is −8.” We may see more than one use of − in the same expression, such as −6 − (−9), where −9 is subtracted from −6. The meaning of the symbol depends on its position in the algebraic expression. Slide 1.5-16

Use the rules for order of operations with real numbers. Objective 4 Use the rules for order of operations with real numbers. Slide 1.5-17

Adding and Subtracting with Grouping Symbols EXAMPLE 7 Adding and Subtracting with Grouping Symbols Perform each indicated operation. Solution: Slide 1.5-18

Perform each indicated operation. EXAMPLE 7 Adding and Subtracting with Grouping Symbols (cont’d) Perform each indicated operation. Solution: Slide 1.5-19

Translate words and phrases involving addition and subtraction. Objective 5 Translate words and phrases involving addition and subtraction. Slide 1.5-20

Translate words and phrases involving addition and subtraction. The word sum indicates addition. The table lists other words and phrases that indicate addition in problem solving. Slide 1.5-21

Translating Words and Phrases (Addition) EXAMPLE 8 Translating Words and Phrases (Addition) Write a numerical expression for the phrase and simplify the expression. 7 increased by the sum of 8 and −3 Solution: Slide 1.5-22

Translate words and phrases involving addition and subtraction Translate words and phrases involving addition and subtraction. (cont’d) The word difference indicates subtraction. Other words and phrases that indicate subtraction in problem solving are given in the table. In subtracting two numbers, be careful to write them in the correct order, because in general, . For example, . Think carefully before interpreting an expression involving subtraction. Slide 1.5-23

Translating Words and Phrases (Subtraction) EXAMPLE 9 Translating Words and Phrases (Subtraction) Write a numerical expression for each phrase, and simplify the expression. The difference between −5 and −12 −2 subtracted from the sum of 4 and −4 Solution: Slide 1.5-24

Solving a Problem Involving Subtraction EXAMPLE 10 Solving a Problem Involving Subtraction The highest Fahrenheit temperature ever recorded in Barrow, Alaska, was 79°F, while the lowest was −56°F. What is difference between these highest and lowest temperatures? (Source: World Almanac and Book of Facts.) Solution: °° Slide 1.5-25

Use signed numbers to interpret data. Objective 6 Use signed numbers to interpret data. Slide 1.5-26

Using a Signed Number to Interpret Data EXAMPLE 11 Using a Signed Number to Interpret Data Refer to Figure 17 and use a signed number to represent the change in the CPI from 2002 to 2003. Solution: Slide 1.5-27