1. Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross Set 4 By Herb I. Gross and Richard A. Medeiros next

2. Solve for n if: n = - 7 – ( - 13 – + 8) Problem #1(a) © 2007 Herbert I. Gross Answer: + 14 next

3. Answer: + 14 Solution: By the “add the opposite” rule - 13 – + 8 = - 13 + - 8 and by our rule for adding two numbers with the same sign we see that - 13 + - 8 = - ( 13 + 8) or - 21. Since we do what's inside the parentheses first, we replace ( - 13 – + 8) by - 21 and rewrite n = - 7 – ( - 13 – + 8) in the equivalent form: - 7 – - 21 (1) next © 2007 Herbert I. Gross

4. Solution: Then again using the “add the opposite” rule, we may rewrite (1) in the equivalent form. - 7 + + 21 (2) Finally, recalling our rule for adding two numbers that have opposite signs, we see that n = + (21 – 7) = + 14 next © 2007 Herbert I. Gross

5. We developed the rules of arithmetic based on whole numbers. As we expanded the number system we made sure that the rules remained intact. Thus, for example the definition of subtraction remains the same regardless of the signs of the numbers. next © 2007 Herbert I. Gross Note 1a

6. In this context - 7 – ( - 13 – + 8) still means the number we must add to - 13 – + 8 to obtain - 7 as the sum. So to check our answer we need only compute the value of ( - 13 + - 8) + + 14 and check that it is equal to - 7. To this end, we may use our rules to show that… next © 2007 Herbert I. Gross Note 1a ( - 13 – + 8 ) + + 14 = + -8-8 - 21 -7-7 ( - 13 – + 8 ) + + 14 = We subtract the smaller magnitude (14) from the greater (21) and keep the sign of the number with the greater magnitude ( - 21) and obtain -7. next

7. The “add the opposite rule” for subtraction is convenient to know, but it is not necessary to memorize it. For example, - 13 – + 8 means the number we must add to + 8 to obtain - 21. We know that if we add - 8 to + 8, the sum will be 0, and if we add - 13 to 0 the sum will be -13. In effect this is the derivation of the add the opposite rule. More specifically we have just illustrated that ( - 13 + + 8) + + 8 = - 13 next © 2007 Herbert I. Gross Note 1a

8. next Solve for n if: n = ( - 7 – - 13) – + 8 Problem #1(b) © 2007 Herbert I. Gross Answer: n = - 2 next

9. Answer: - 2 Solution: We still work within the parentheses first: ( - 7 – - 13) – + 8 next © 2007 Herbert I. Gross ( - 7 – - 13) = ( - 7 + + 13) +6+6 – + 8 (1)(1) next

10. Solution: By the “add the opposite” rule we may replace (1) + 6 – + 8 by… n = + 6 + - 8 (2) next © 2007 Herbert I. Gross By our rule for adding two numbers that have different signs, we may replace (2) by: n = - (8 – 6) = - 2 next

11. Notice the need here for grouping symbols or at least a rule for the order of operations. Namely if we omit the parentheses, - 7 – ( - 13 – + 8) and ( - 7 – - 13) – + 8 would both look like - 7 – - 13 – + 8. next © 2007 Herbert I. Gross Note 1b

12. Therefore in order for our PEMDAS agreement to remain in effect, we would have to interpret - 7 – - 13 – + 8 as ( - 7 – - 13) – +8. By way of review the PEMDAS agreement tells us that when we have an ambiguous expression that involves only addition and/or subtraction, we do the arithmetic from left to right next © 2007 Herbert I. Gross Note 1b

13. next Solve for n if: - n = + 7 – - 4 Problem #2 © 2007 Herbert I. Gross Answer: n = - 11 next

14. Answer: - 11 Solution: By the “add the opposite” rule we know that + 7 – - 4 = + 7 + + 4 and by our rule for adding two numbers with the same sign we see that… + 7 + + 4 = + 11. Hence we may rewrite - n = + 7 – - 4 in the equivalent form: - n = - 11 However, we are asked to find the value of n rather than the value of -n. next © 2007 Herbert I. Gross

15. Solution: However, we are asked to find the value of n rather than the value of -n. Since n is the opposite of - n and since - n = + 11, n must be the opposite of + 11. The opposite of + 11 is - 11. Therefore n = - 11 next © 2007 Herbert I. Gross

16. Sometimes what seems “logical” can be incorrect. For example, you might think that taking the opposite of a number means to take the opposite of every sign. Thus you might be inclined to replace every sign by its opposite to convert + 7 – - 4 into - 7 + + 4; and thus obtain that n = - 3 next © 2007 Herbert I. Gross Note 2

17. The point is that we are bound by the properties and rules that we have accepted; not by what “seems” true. In the present exercise our rules tell us that… - n = + 7 – - 4 means the same thing as… - n = + 7 + - 4 = + 11 and the opposite of + 11 = - 11. next © 2007 Herbert I. Gross Note 2

18. Reading comprehension is important. While one might feel that a mistake in sign is minor, in general there is a big difference between n and - n. For example suppose you computed that n = 498, and you then wanted to evaluate the expression 502 + n. We see that the answer would be 502 + 498 = 1000. next © 2007 Herbert I. Gross Note 2

19. However suppose you had made a mistake in sign and that the correct value of n was - 498. In that case we would have obtained 502 + - 498 or 4, and the difference between 4 and 1,000 is significant. next © 2007 Herbert I. Gross Note 2

20. next a. Which number is greater, and by how much? - 6 – 4 or 4 – - 6 Problem #3(a) © 2007 Herbert I. Gross Answer: 4 – - 6 by 20 next

21. Answer: - 11 Solution: When there is no sign we assume the number is positive. Hence, - 6 – 4 means the same thing as - 6 – + 4. By our “add the opposite” rule we see that… - 6 – + 4 = - 6 + - 4 = - (6 + 4) = - 10. On the other hand… + 4 – - 6 = + 4 + + 6 = + 10. Since + 10 – - 10 = + 10 + + 10 = + 20, we see that + 4 – - 6 exceeds - 6 – - 4 by 20. next © 2007 Herbert I. Gross

22. This exercise emphasizes the fact that when we change the order of the two numbers in a subtraction problem we change the sign of the answer but not the magnitude. next © 2007 Herbert I. Gross Note 3a

23. In terms of the number line, - 6 – + 4 measures the directed distance in going from the point + 4 to the point - 6, while + 4 – - 6 measures the same distance but in the direction from the point - 6 to the point + 4. next © 2007 Herbert I. Gross Note 3a

24. In terms of profit and loss it indicates the difference between converting a $4 profit into a $6 loss and converting a $4 loss into a $6 profit. next © 2007 Herbert I. Gross Note 3a In terms of temperature change it indicates the difference between the temperature going from 4 degrees above 0 to 6 degrees below 0 and going from 4 degrees below 0 to 6 degrees above 0. next

25. b. Which number is greater, and by how much? │ - 6 – 4│ or │ 4 – - 6│ Problem #3(b) © 2007 Herbert I. Gross next Answer: They are equal.

26. Solution: In exercise 3a, we saw that - 6 – + 4 = - 10, and + 4 – - 6 = + 10. The signs of - 10 and + 10 are different, but their magnitudes are the same. This means that │ - 6 – + 4│ = │ + 4 – - 6│. next © 2007 Herbert I. Gross

27. There is a tendency to think of an expression such as - 6 – + 4 as being made up of two numbers. However while it looks a bit more complicated it is still just one number; namely, - 10. So the more complicated looking equality… | - 6 – + 4| = | + 4 – - 6| is simply another way of writing that… | - 10| = | + 10|. next © 2007 Herbert I. Gross Note 3b

28. In terms of more visual examples, converting a $4 loss into a $6 profit is a different transaction from converting a $4 profit into a $6 loss. However, the size of the transaction in either case is $10. next © 2007 Herbert I. Gross Note 3b Or in terms of directed distance it is a movement of 10 units whether we go from - 4 to + 6 or from + 6 to - 4. next

29. In terms of temperature change it is a 10 degree change in temperature whether the temperature decreased from 4 above 0 to 6 below 0, or it increased from 6 degrees below 0 to 4 degrees above 0 next © 2007 Herbert I. Gross Note 3b 0 +8+8 +7+7 +6+6 +5+5 +4+4 +3+3 +2+2 +1+1 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7 -8-8 1 2 3 4 5 6 7 8 9 10 0 +8+8 +7+7 +6+6 +5+5 +4+4 +3+3 +2+2 +1+1 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7 -8-8 1 2 3 4 5 6 7 8 9 10 degree change next

30. Evaluate: 8 – (6 – n) when n = 9. Problem #4 (a) © 2007 Herbert I. Gross Answer: 11 next

31. Answer: 11 Solution: If we replace n by 9 in the expression 8 – (6 – n), we obtain the number 8 – (6 – 9). We know that (6 – 9) = (6 + - 9) = - (9 – 6) = - 3. Therefore... 8 – (6 – 9) = 8 – - 3 = 8 + + 3 = + 11. next © 2007 Herbert I. Gross

32. Once we think we have the correct answer we should be able to check it just by looking at the basic definition of subtraction. In this case we are looking for the number that must be added to 6 – 9 to obtain 8 as the sum. In other words, we are looking for the number which must be added to - 3 to yield 8 as the sum. So as a check we see that - 3 + 11 = 8. next © 2007 Herbert I. Gross Note 4a

33. A more difficult version of this exercise is given in Exercise 4(c). Namely once we know that n = 9, it is relatively easy to compute the value of 8 – (6 – n). However, it is not quite as simple if we are asked to rewrite 8 – (6 – n) in an equivalent from that is free of grouping symbols. next © 2007 Herbert I. Gross Note 4a

34. We'll look at the above note in greater detail in our solution to problem 4(b) but, as a prelude let's first look at an alternative way to find the number we must add to 6 – 9 to obtain 8 as the sum.. next © 2007 Herbert I. Gross Note 4a

35. To this end, we may first rewrite 6 – 9 as 6 + - 9. If we add - 6 + + 9 to 6 + - 9, the sum becomes 0, and if we then add + 8 to 0 the sum becomes 8. Thus, in all we added - 6 + + 9 + + 8 or + 11 to 6 – 9 to obtain 8 as the sum. next © 2007 Herbert I. Gross Note 4a

36. next What must we add to (6 – n) to obtain 8 as the sum? Problem #4 (b) © 2007 Herbert I. Gross Answer: n + 2 next

37. Answer: n + 2 Solution: This is a paraphrase of our last note above, with n replacing 9. More specifically, 8 – (6 – n) means the number we must add to 6 – n to obtain 8 as the sum. (6 – n) can be written in the equivalent form (6 + - n). next © 2007 Herbert I. Gross

38. Solution: To convert 6 + - n to 8 using addition, we may add 2 to 6 to obtain 8; and we can add n to - n to obtain 0. In other words, using our rules for arithmetic, we see that… next © 2007 Herbert I. Gross (6 + - n) + (n + 2) = (6 + 2) + ( - n + + n) = = 8 + 0 8 In summary, n + 2 is the number we must add to 6 – n to obtain 8 as the sum. next

39. How are the expressions 8 – (6 – n) and n + 2 related? Problem #4 (c) © 2007 Herbert I. Gross Answer: They are equal. next

40. Answer: They are equivalent. Solution: In part (b) we showed that n + 2 is the number we must add to (6 – n) to obtain 8 as the sum; and that by definition is what 8 – (6 – n) means. next © 2007 Herbert I. Gross

41. In a later presentation we will show a more efficient way to show that 8 – (6 – n) = n + 2. For now it is important to see how the power of paraphrasing can simplify a problem. Namely what we have shown is that Program #2 below is a simpler version of Program #1.. next © 2007 Herbert I. Gross Note 4c

42. next © 2007 Herbert I. Gross Note 4c Start with any number. n 12 Start with any number. n Subtract it from 6. (6 – n) -6-6 Add 2. ( n + 2) Subtract it from 8. (8 – - 6) 14 Write your answer. 14 Write your answer. Program #1Program #2 12 14 12)(12 next

43. For what value of n will: 8 – (6 – n) = 23 Problem #4 (d) © 2007 Herbert I. Gross Answer: 21

44. Solution: In exercise (c) we saw that 8 – (6 – n) is equivalent to n + 2. This means that in any equation we can replace 8 – (6 – n) by n + 2. In other words the equation 8 – (6 – n) = 23 is equivalent to the equation n + 2 = 23. And if we now subtract 2 from both sides of the equation we see that n = 21. next © 2007 Herbert I. Gross

45. As a check we may replace n by 21 in the expression … 8 – (6 – n) next © 2007 Herbert I. Gross Note 4d 21) to obtain… 8 – (6 – 21) = 8 – - 15 = 8 + 15 = 23 Thus verifying that n = 21 is the solution for the equation… 8 – (6 – n) = 23 next

46. You are hiking along a trail. The positive direction is the one in which your altitude (measured in feet and labeled “above sea level” and “below sea level”) is increasing, and the negative direction is the one in which your altitude is decreasing. Make up a word problem for which the correct answer is found by computing the value of 300 – - 250. Problem #5 © 2007 Herbert I. Gross

47. Answer: One form of the answer is… By how much has your elevation increased if you go from 250 feet below sea level to 300 feet above sea level? Solution: By definition 300 – - 250 is the directed distance in going from - 250 (that is, from 250 feet below sea level) to + 300 (that is, to 300 feet above sea level). next © 2007 Herbert I. Gross

48. There is a big difference between being able to perform an operation and knowing when to use the operation in a “real world” situation. With respect to this exercise knowing how to subtract two signed numbers by using, say, the “add the opposite” rule doesn't guarantee that a person will recognize that subtraction measures the (directed) gap between two numbers. next © 2007 Herbert I. Gross Note 4c