('')''* = 1- $302. It is common to include parentheses around negative numbers when they appear after an operation symbol.

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1 2.2 ADDING INTEGERS Adding Integers with the Same Sign We often associate the + and - symbols with positive and negative situations. We can find the sum of integers by considering the outcome of these situations as illustrated below. A price decrease of $1 ('')''* followed (')'* by a price decrease ('')''* of $2 results ()* T T T 1- $ $22 in a decrease of $. ('')''* T T = 1- $2 In the same sense, an increase followed by an increase results in an increase, or a positive outcome. EXAMPLE 1 (a) Fill in the blank. A loss of $1 followed by a loss of $1 results in a. Write the math symbols that represent the situation described in (a). Student Learning Objectives After studying this section, you will be able to: Add integers with the same sign. Add integers with different signs. Evaluate algebraic expressions involving addition of integers. Solve applied problems involving addition of integers. (a) A loss of $1 followed by a loss of $1 results in a loss of $2. A loss of $1 followed by a loss of $1 results in a loss of $2. $1 1$12 $2 It is common to include parentheses around negative numbers when they appear after an operation symbol. Practice Problem 1 (a) Fill in the blank. A decrease in altitude of 1 feet followed by a decrease in altitude of 1 feet results in a. Write the math symbols that represent the situation described in (a). Since integers are often used to indicate direction and distance, we can also use the number line to find the sum of numbers such as A move to the right on the number line is a move in the positive direction, and a move to the left on the number line is a move in the negative direction. The direction we move on the number line is indicated by the sign of the number. We see that 1+( 2)=. We start at and move 1 unit in the negative direction, followed by another 2 units in the negative direction negative direction positive direction 11

2 114 Chapter 2 Integers EXAMPLE 2 (a) Begin at on the number line and move units to the left followed by another 2 units to the left Is the end result in the positive or the negative region? (c) Write the math symbols that represent the situation. (d) Use the number line to find the sum. (a) From the illustration we see that the end result is in the negative region since we began at and moved units in the negative direction (left), followed by another 2 units in the negative direction. left followed by left 2 (c) The math symbols are: ( 2) (d) We end at -5, which is the sum = -5 Practice Problem 2 (a) Begin at on the number line and move 4 units to the left followed by another 1 unit to the left Is the end result in the positive or the negative region? (c) Write the math symbols that represent the situation. (d) Use the number line to find the sum. Example 2 shows that if we make a move in the negative direction followed by another move in the negative direction, the result is in the negative region. When we add negative numbers, we are repeatedly moving in the negative direction and thus the sum is a negative number. How do we find the sum of numbers with the same sign without a number line? Let s look at the results from Example = - We are adding two negative numbers, and the sum is negative = - 5 We must add the absolute values to get 5: 2 + = 5. Of course, we know that when we add two positive numbers the answer is a positive number. We state the formal rule.

3 Section 2.2 Adding Integers 115 RULE FOR ADDING TWO OR MORE NUMBERS WITH THE SAME SIGN To add numbers with the same sign: 1. Use the common sign in the answer. 2. Add the absolute values of the numbers. If all the numbers are positive, the sum is positive. If all the numbers are negative, the sum is negative. EXAMPLE We are adding two numbers with the same sign, so we keep the common sign (negative sign) and add the absolute values = = - 4 Practice Problem The answer is negative since the common sign is negative. 1 + = () 4 Adding Integers with Different Signs So far we have seen how to add numbers with the same sign. We use a similar approach to see how we add numbers that have different signs. Addition of numbers with different signs often involves situations such as a decrease followed by an increase or a quantity that rises and then falls. If we wish, we can also use a vertical number line instead of a horizontal number line to illustrate these types of situations. If we move up, we are moving in the positive direction. If we move down, we are moving in the negative direction. EXAMPLE 4 One night the temperature on Long Island, New York, dropped to -25 F. At dawn the temperature had risen 1 F. (a) Write the math symbols that represent the situation. Use the thermometer to determine if the temperature at dawn was positive or negative. (c) Find the sum F down 25 followed by up 1 T T T 4 (a) - 25 F F2 From the chart we see that the temperature reading was negative since it went down 1-2 more degrees than it went up 1+2. (c) The final temperature is -15 F, which is the sum: -25 F + 1 F = -15 F. 1 25F Practice Problem 4 Last night the temperature in Boston, Massachusetts, dropped to -15 F. At dawn it had risen F. (a) Write the math symbols that represent the situation. Use the thermometer to determine if the temperature at dawn was positive or negative. (c) Find the sum

4 116 Chapter 2 Integers Example 4 involves addition of integers with different signs. How do we perform this addition without using a chart? Let s look at the results from Example = - The sign of the sum is negative since we move a larger distance in the negative direction = -15 We must subtract 25-1 to get the result, 15. We see that to find the sum, we actually find the difference between 25 and 1. Also notice that if we do not account for the sign, the larger number is 25. The sign of 25 is negative, and the answer is also negative. This suggests the addition rule for two numbers with different signs. RULE FOR ADDING TWO NUMBERS WITH DIFFERENT SIGNS To add two numbers with different signs: 1. Use the sign of the number with the larger absolute value in the answer. 2. Subtract the absolute values of the numbers. In other words, we keep the sign of the larger absolute value and subtract We move a larger distance in the negative direction, and the answer is negative. We move a larger distance in the positive direction, and the answer is positive. EXAMPLE 5 (a) We are adding numbers with different signs, so we keep the sign of the larger absolute value and subtract. (a) = - The answer is negative since - is negative and has the larger absolute value = - 1 Subtract: - 2 = = + The answer is positive since is positive and has the larger absolute value = + 1 Subtract: - 2 = 1. Practice Problem 5 (a) We summarize the rules for adding numbers as follows. Adding numbers with the same sign: Keep the common sign and add the absolute values. Adding numbers with different signs: Keep the sign of the larger absolute value and subtract the absolute values EXAMPLE 6 (a) (a) = (- 5 ) = + or = = -1 We have different signs. 8 is larger than 5, so the answer is positive. Subtract: 8-5 =. We have the same sign. Keep the common sign. Add: = 1.

5 Section 2.2 Adding Integers 117 Practice Problem 6 (a) Recall from Section 2.1 that the numbers and - are called opposites. Let s look at the number line in the margin and see what happens when we add the opposites When we add opposites the sum is because we move the same distance in the positive direction as we move in the negative direction. This fact is referred to as the additive inverse property, and thus and - are also called additive inverses. 2 1 ADDITIVE INVERSE PROPERTY 1 For any number a, a + 1-a2 = and 1-a2 + a = The sum of any number and its opposite is. EXAMPLE 7 (a) Find x. x + 21 = (a) Since 1298 and are additive inverses, their sum is = The sum of additive inverses is. Thus if x + 21 =, then x = -21 since =. Practice Problem 7 (a) Find y y = If there are three or more numbers to add, it may be easier to add positive numbers and negative numbers separately and then combine the results. We can do this because, just like with whole numbers, addition of integers is commutative and associative. EXAMPLE = (9 + 12) = -7 + (9 + 12) = = 14 Change the order of addition and regroup. Add the negative numbers: = -7. Add the positive numbers: = 21. Add the result: = 14. Practice Problem

6 118 Chapter 2 Integers Evaluating Algebraic Expressions Involving Addition of Integers We evaluate expressions involving integers just as we did expressions involving whole numbers. We replace the variables with the given numbers and perform the operations indicated. EXAMPLE 9 Evaluate. (a) -7 + a + b for a = - and b = 9 -x + y for x = -2 and y = -6 We place parentheses around the variables, and then replace each variable with the appropriate values. (a) a2 + 1b2-1x2 + 1y = = = -1 = = = -17 Practice Problem 9 Evaluate. (a) - + x + y for x = -5 and y = -11 -x y for x = -2 and y = -15 Solving Applied Problems Involving Addition of Integers EXAMPLE 1 Information on Micro Firm Computer Sales profit and loss situation is given on the graph. What was the company s overall profit or loss at the end of the third quarter? Profit or loss (thousands of dollars) st 2nd rd 4th Quarter 1st quarter loss 2nd quarter profit rd quarter profit net profit $4, $2, $6, $4, 4,+(2,+6,)= 4,+8,=4, At the end of the third quarter the company had a net profit of $4,. Practice Problem 1 What was Micro Firm Computer Sales overall profit or loss at the end of the second quarter?

7 Section 2.2 Adding Integers 119 When to Use a Calculator A calculator is an important tool and therefore we benefit by learning how to use it. You may be thinking, Why learn math if I can use a calculator? Well, often it is not practical or convenient to use a calculator. Many times we must perform calculations unexpectedly and do not have a calculator available, as in the following situations. You receive change for a purchase made. Is the change correct? You are shopping and notice that an item you d like to buy is marked down percent. You must calculate the reduced price to determine if you can afford to buy the item. You have lunch with four friends and the bill is on one check. How much do you owe for your lunch? These are just a few of the many situations that require you to use your knowledge of mathematics. Besides, even if you had a calculator in the situations above, you would still need to know how to go about solving the problem. Should you add, subtract, multiply, or divide? The calculator does only what you tell it to do; it does not plan the approach! Learning how to do mathematics develops problem-solving skills, and the calculator assists us in solving problems. Calculator Adding Negative Numbers There are a few different ways to enter negative numbers in a calculator. Usually, either a or the 1-2 key is used. You should read the manual for directions. To find , enter 119 +> = The display should read -4. +> - Exercises Do the following exercises without and then with a calculator. Which way is faster, with or without a calculator? Combine like terms. 4xy + 2x + xy. If you didn t know the rules for combining like terms, could you have done exercise 2 with a calculator?