209 CH 39 ADDING AND SUBTRACTING SIGNED NUMBERS Introduction Now that we know a little something about positive and negative (signed) numbers, and we re pretty good at the Jeopardy scoring system, we need to analyze what happens when we add and subtract such numbers. Homework 1. Do each addition or subtraction problem -- think Jeopardy: a. 17 + 3 b. 16 5 c. 7 8 d. 12 12 e. 23 30 f. 2 + 10 g. 3 + 2 h. 12 3 i. 80 100 j. 30 + 40 k. 20 + 5 l. 80 20 m. 0 + 32 n. 0 32 o. 88 + 0 p. 34 0 q. 10 15 r. 23 + 80 s. 7 + 2 t. 18 18 u. 9 12 v. 5 + 9 w. 10 3 x. 10 + 2 y. 30 4 z. 3 100 The Law of Adjacent Signs We should now be adept at adding and subtracting signed numbers when there is one sign (either addition or subtraction) between the numbers. But often there are two signs -- right next to each other --
210 between the numbers (we say that the signs are adjacent). For example, what will we do if we come across problems like 18 + (+3) 10 ( 2) 7 + ( 2) 8 (+5)? It s easy. The Law of Adjacent Signs (a term invented by the author) tells us that if the two adjacent signs are the same, replace them with a single addition sign; and if the two adjacent signs are different, replace them with a single subtraction sign. Then apply the Jeopardy ideas from the previous section and you re done. (A justification of the Law of Adjacent Signs is covered later in this chapter.) Thus, in the four problems above, 18 + (+3) = 18 + 3 = 21 (same sign -- change to addition) 10 ( 2) = 10 + 2 = 12 (same sign -- change to addition) 7 + ( 2) = 7 2 = 5 (different signs -- change to subtraction) 8 (+5) = 8 5 = 13 (different signs -- change to subtraction) EXAMPLE 1: A. With our Jeopardy rules and the Law of Adjacent Signs, we can now work out any addition or subtraction problem involving signed numbers. For example, consider the problem 10 + ( 3). By the Law of Adjacent Signs, we can change the problem to 10 3, which, of course, equals 7. B. 12 (+4) = 12 4 = 8 C. 3 + ( 4) = 3 4 = 1 D. 8 (+9) = 8 9 = 1 E. 6 ( 2) = 6 + 2 = 8
211 EXAMPLE 2: A. 8 + ( 3) = 8 3 = 11 B. 7 (+5) = 7 5 = 12 C. 9 + (+4) = 9 + 4 = 5 D. 12 + (+17) = 12 + 17 = 5 E. 3 ( 4) = 3 + 4 = 1 F. 7 ( 2) = 7 + 2 = 5 EXAMPLE 3: A. 7 + ( 3) + 8 B. 13 (+3) + ( 5) = 7 3 + 8 = 13 3 5 = 4 + 8 = 16 5 = 12 = 21 C. 8 ( 1) ( 3) D. 13 + ( 7) ( 1) = 8 + 1 + 3 = 13 7 + 1 = 7 + 3 = 6 + 1 = 4 = 7 E. 12 + ( 3) (+9) ( 2) + (+7) + 8 12 = 12 3 9 + 2 + 7 + 8 12 (use the Law of Adjacent Signs) = 15 9 + 2 + 7 + 8 12 ( 12 3 = 15) = 24 + 2 + 7 + 8 12 ( 15 9 = 24) = 22 + 7 + 8 12 ( 24 + 2 = 22) = 15 + 8 12 ( 22 + 7 = 15) = 7 12 ( 15 + 8 = 7) = 19 (and we re finally done!)
212 Opposites Every number has an opposite. The opposite of 17 is 17, and the opposite of 3.4 is 3.4. The opposite of 0 is 0, which might seem a bit strange; challenge your instructor to prove it to you. Sometimes we come across an expression like ( 4). Since the signs are adjacent to each other, and they re the same sign, we can use the Law of Adjacent Signs to combine them into a single plus sign. Thus, ( 4) = +4 = 4 This double negative idea can be used to solve a problem like this: ( 7) + ( 5) = 7 5 = 2 What happens if we add a number and its opposite together? If we add 7 and its opposite, we get 7 + ( 7) = 7 7 = 0. If we add 13 and its opposite, we get the same sum: 13 + (+13) = 13 + 13 = 0 It seems that The sum of a number and its opposite is zero. By the way, this rule is the basis for proving that the opposite of 0 is 0.
213 Homework 2. Find the value of each expression (that is, simplify): a. 17 + 3 b. 8 + 3 c. 8 + ( 1) d. 7 + ( 3) e. 3 + ( 10) f. 5 + ( 13) g. 9 + 0 h. 7 + 12 i. 20 + 10 j. 30 + ( 2) k. 0 + ( 3) l. 10 + 10 m. 12 + 7 n. 3 + ( 20) o. 9 + ( 9) p. 1 + ( 12) 3. Find the value of each expression: a. 6 + ( 6) + ( 1) b. 4 + 8 + ( 8) + ( 7) c. 3 + ( 4) + 5 d. 5 + ( 5) + ( 3) e. 12 + ( 3) + ( 12) f. ( 1) + ( 2) + ( 3) g. 2 + ( 2) + ( 3) + 3 h. 8 + (7) + 8 + ( 1) 4. Find the value of each expression: a. 10 2 b. 2 10 c. 3 8 d. 4 2 e. 5 ( 1) f. 10 ( 3) g. 3 ( 3) h. 1 ( 7) i. 18 22 j. 7 5 k. 23 ( 4) l. 8 ( 2) m. 7 20 n. 99 1 o. 1 ( 2) p. 12 ( 4) q. 13 21 r. 4 9 s. 12 ( 9) t. 5 ( 1) 5. Find the value of each expression: a. 7 + ( 3) 8 b. 3 + 0 ( 1) c. 8 2 1 d. 2 ( 3) + 1 e. 7 ( 3) 8 f. 9 2 3 4 g. 2 + 5 + 9 1 h. 1 ( 1) ( 1) i. 8 + ( 2) 8 ( 1) j. ( 3) + 7 k. ( 1) 3 l. ( 5) ( 4) m. ( 11) ( 8) n. ( 7) (+5) o. ( 3) + ( 3)
214 A Justification of the Law of Adjacent Signs 1. 5 + (+3) = 5 + 3 = 8 Since +3 = 3, we don t need the parentheses, and so the two plus signs turn into a single addition sign. 2. 10 + ( 2) = 10 2 = 8 10 is a gain -- 2 is a loss; combining a gain of 10 with a loss of 2 produces a net gain of 8. Thus, the two different signs turn into a single subtraction sign. 3. 20 (+5) = 20 5 = 15 As in part 1, the plus sign on the 5 is redundant (not needed). 4. 20 ( 5) = 20 + 5 = 25 This is the weirdest situation. Suppose you currently have $25 in Jeopardy and miss a $5 question -- that puts you at $20 (the first number in the problem). Right after the commercial Alex informs you that the judges have decided that the $5 question you missed before the break was bogus and will be thrown out. So, the $5 loss must be removed from your score of $20: 20 ( 5) = 20 + 5 = 25 removing a $5 loss = getting the $5 back which puts your score back to the $25 you had before the bogus $5 question.
215 Alternate Approach: Using The Number Line 5 4 3 2 1 0 1 2 3 4 5 EXAMPLE 2: A. 3 + 2 = 5 Start at 3 on the number line; move 2 units to the right, and you end up at 5. B. 4 + 5 = 1 Start at 4; move 5 units to the right, and you end up at 1. C. 5 + 3 = 2 Start at 5; move 3 units to the right, and you end up at 2. D. 5 3 = 2 Start at 5; move 3 units to the left, and you end up at 2. E. 3 7 = 4 Start at 3; move 7 units to the left, and you end up at 4. F. 3 2 = 5 Start at 3; move 2 units to the left, and you end up at 5. All of these results could have been obtained by thinking about Jeopardy or using the Law of Adjacent Signs. You may pick any method you like to add and subtract positive numbers -- just be sure you can do it really well. Now for some problems where we add and subtract negative numbers.
216 Practice Problems 6. Evaluate each expression: a. 7 ( 3) ( 12) b. ( 8) 10 + 3 + ( 10) c. 8 ( 2) ( 9) d. 5 + ( 8) + 4 11 e. 5 + ( 11) + 1 + ( 8) f. 3 ( 2) ( 11) 7 3 g. ( 1) 7 11 + ( 5) h. 4 ( 7) + 12 i. ( 2) ( 2) + ( 7) ( 7) 4 j. 11 + 6 + ( 6) + 6 k. 6 + 9 12 + 11 l. 5 ( 6) + ( 4) 7 ( 3) m. 2 + 2 + 9 ( 6) + 8 n. 6 6 1 o. 6 4 7 ( 10) p. 2 ( 12) + ( 7) q. ( 9) + 8 ( 5) + 9 r. 6 + 2 8 + 4 + 7 s. 9 ( 5) + 10 + ( 11) t. 12 + 9 2 ( 5) ( 1) u. 9 + 9 5 12 v. 2 11 3 w. ( 9) + 3 2 ( 10) x. 8 + 6 + ( 1) y. ( 12) + ( 9) ( 1) 3 + ( 7) z. ( 7) ( 5) + ( 9) 4 Solutions 1. a. 20 b. 11 c. 1 d. 0 e. 7 f. 8 g. 1 h. 15 i. 20 j. 10 k. 15 l. 100 m. 32 n. 32 o. 88 p. 34 q. 25 r. 57 s. 5 t. 36 u. 3 v. 4 w. 13 x. 8 y. 34 z. 97
217 2. a. 20 b. 5 c. 9 d. 4 e. 13 f. 8 g. 9 h. 5 i. 10 j. 32 k. 3 l. 0 m. 5 n. 23 o. 0 p. 11 3. a. 1 b. 3 c. 2 d. 13 e. 3 f. 6 g. 0 h. 22 4. a. 8 b. 8 c. 11 d. 6 e. 6 f. 13 g. 0 h. 6 i. 4 j. 12 k. 19 l. 10 m. 13 n. 100 o. 1 p. 16 q. 8 r. 13 s. 21 t. 4 5. a. 4 b. 2 c. 11 d. 2 e. 2 f. 0 g. 11 h. 1 i. 1 j. 10 k. 2 l. 9 m. 19 n. 2 o. 0 6. a. 8 b. 9 c. 19 d. 20 e. 13 f. 0 g. 22 h. 15 i. 0 j. 17 k. 2 l. 3 m. 23 n. 13 o. 5 p. 7 q. 31 r. 11 s. 5 t. 1 u. 17 v. 16 w. 20 x. 3 y. 6 z. 1
218 Good people are good because they've come to wisdom through failure. William Saroyan