Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding and Subtracting Real Numbers Add two numbers with the same sign. Add positive and negative numbers. Use the definition of subtraction. Use the rules for order of operations with real numbers. Interpret words and phrases involving addition and subtraction. Use signed numbers to interpret data. 6 6

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Add two numbers with the same sign.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To add two numbers with the same sign, add the absolute values of the numbers. The sum has the same sign as the given numbers. Example: 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 sum of the sum of their absolute values. To avoid confusion, two operation symbols should not be written successively without a parenthesis between them.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Use a number line to find each sum. Adding Numbers on a Number Line Solution: 5 −7−7

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Find the sum. Solution: Adding Two Negative Numbers

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Add positive and negative numbers.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Add two numbers with the different signs. For instance, to add −12 and 5, find their absolute values: and Then find the difference between these absolute values: The sum will be negative, since, so the final answer is. To add two numbers with different signs, find the absolute values of the numbers and subtract the smaller absolute value from the larger. Give the answer the sign of the number having the larger absolute value.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Adding Numbers with Different Signs Use a number line to find the sum. 3

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Adding Mentally Correct Check each answer. Correct

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Use the definition of subtraction.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley We can illustrate the subtraction of 4 from 7, written, with a number line. Use the definition of subtraction. The procedure to find the difference is exactly the same procedure that would be used to find the sum, so

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use the definition of subtraction. (cont’d) The previous equation suggests that subtracting a positive number from a larger positive number is the same as adding the additive inverse of the smaller number to the larger. From this comes the definition of subtraction, for any real numbers x and y, That is, to subtract y from x, add the additive inverse (or opposite) of y to x.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solution: Use the Definition of Subtraction Subtract.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.” Uses of the Symbol − We may see more than one use of − in the same problem, such as −6 − (−9) where −9 is subtracted from −6. The meaning of the symbol depends on its position in the algebraic expression.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Use the rules for order of operations with real numbers.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Perform the indicated operations. EXAMPLE 6 Adding and Subtracting with Grouping Symbols Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Interpret words and phrases involving addition and subtraction.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Interpret words and phrases involving addition. The word sum indicates addition. The table lists other words and phrases that indicate addition in problem solving.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solution: Write a numerical expression for the phrase and simplify the expression. 7 increased by the sum of 8 and −3 Interpreting Words and Phrases Involving Addition

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Interpret words and phrases involving subtraction. 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.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Write a numerical expression for the phrase and simplify the expression. −2 subtracted from the sum of 4 and −4 Solution: Interpreting Words and Phrases Involving Subtraction

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 2006.) EXAMPLE 9 Solving a Problem Involving Subtraction Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6 Objective 6 Use signed numbers to interpret data.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution: EXAMPLE 10 Using a Signed Number to Interpret Data Refer to Figure 17 and use a signed number to represent the change in the PPI from 2002 to 2003.