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ORAL EXERCISES In each of the following exercises state first the sign of the result, then give the complete answer. 1. 25 -5.

18. (-12 a) =(-4). 2. (-27) -9.

19. (-12 b) =(-6). 3. (-27) + (-9).

20. (-8 xy) -8. 4. 27 =(-9).

21. (-144 r) =(-12). 5. (-25) =(-5).

22. 16 abc-4. 6. 30=(-6).

23. 1:(-). 7. (72) = 12.

24. (-)-(-1). 8. (-60) =(-4).

25. (–2.5) = (2.5). 9. (-10) = 10.

26. 7.26 =(-2.42). 10. (-8) +8.

27. (-a) +6; - Ans. 11. 90 =(-15). 12. (-63) =(-7).

28. a :(-6). 13. (-100) = 25.

29. (-a)=(-6). 14. 121 : 11.

30. -n. 15. (–144) =(-12).

31. – 4 x = x. 16. 12 a -4.

32. –4x :(-x). 17. 12 a:(-4).

33. Give (by inspection) the value of x in each of the following equations ;

(a) – x=6.. (e) – 4 x=-16. () –4=2 x. (b) 4 x=-8. (f) –6 x=36. (j) – 10=5 x. (c) –3x=-12. (g) 3 x=-15. (k) = -3. (d) – 5 x=10. (h) – 2 x=13. (1) -2-1

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WRITTEN EXERCISES If a=2, b=-3, and c= -5, find the value of each of the following expressions.

1. 62. 2. 2 abc 3. ab 4. abc. 6. 2 ab 6. 4 be 7. ap. 8. a+b

a-6 ab+acbc 10. 5 a—3b+2 c 11. abc – 3 bc+c.

3 c
aʼ+62 +62

V2 a-bc

Vbc+1 18 a-bb-c

ai+b3 + c3

a4+64+c4 a+b+b+c a2+62+c2

a+b+c 1. a? +62 Q4 +64

"bti

b+c 21. Explain how the equation (-4)=(-2)=2 illustrates the following statement: “ The boy who owes $4 owes twice as much as the boy who owes $2.”

22. Taking the equation in Ex. 21, write out the statement it illustrates with reference to the latitudes of two ships. [Hint. See § 16.]

23. Write (using negative numbers whenever necessary) the equation corresponding to the following statement : “If John owes $10 and I owe half as much, I owe $5."

24. Explain how the equation 10:(-2)= -5 illustrates the following statement: “ If I have $10 and you owe $2 then I have five times as much as you but in the opposite

20. abc . a+b

a+b^03 53: 19. abcyath

sense.

For further exercises on this topic, see the review exercises, p. 47, and Appendix, p. 293.

EXERCISES – REVIEW OF CHAPTER II

1. The words “good” and “bad” have precisely opposite meanings; hence the one may be called the negative of the other. Two such words are also called antonyms of each other. State what is the negative (or antonym) of each of the following expressions;

(a) slow
(e) to find anything

(i) darkness (6) rich

(f) to give away something (3) evil (c) difficult (9) to go to sleep

(k) dwarf (d) industrious (h) to stand up

(1) doubt 2. What tense (in grammar) is the negative of the past tense?

3. What word describes the negative of “good health”?

4. If you have any two numbers (positive or negative) how do you decide which is the greater?

(HINT. See $ 20.]

6. In each of the following cases, subtract (mentally) the smaller number from the larger;

(a) 5, 6. (c) 14, –10. (e) }, . (g) .5, -.6

(6) -1, 3. (d) -4, -5. (f) } ,-. (h) - }, .3 6. In each of the examples in Ex. 5 subtract (mentally) the larger number from the smaller.

7. Find (mentally) the value of each of the following expressions; (a) 5X(-6). (d) {X(-}). (g) (-1)3.

-1 (6) (-5) X(-9

(h) (-3)2+2. (c) (-5). (f) }X(-2)X(-). (i) (-3)3 – 10.

8. State the value of each of the following expressions when p = 1.

(a) p+1. (d) 2-p. (9) p(-p). '(j) pb. (6) p-1. (e) 2–2 p. (h) 072 (k) (-p)". (c) p-2. (f) p2–1. (i) V4p2 (1) po–pa+p-1.

9. State the value of each of the parts of Ex. 8 when pr-2.

10. State the value of each of the following expressions when p=1 and q=-2: (a) p+q. (d) 2 p+q. (9) pq+1. (1) 229.

W 4p (6) p-a. (e) 3 p-2q. (h) 2 p?q?.

(k) qe p3. (c) q-p. (f) pa+q?. (c) 2 pq?. (1) 2p+ 4 q. 11. Write xx2x3+2 y2y3 in its simplest form.

12. Find the value of each of the following expressions when x=1 and y=2. (a) 6 xży 9 xy+y3.

(c) 3Vx-4V2 y–2 x2+4 y. (b) { x3 – xy2 —4 y.

(2) 3x2–4y+V4 x

XP+y 13. In arithmetic the least of all numbers used is 0. Is there any such thing as the least of all numbers used in algebra ? Explain.

14. If any number be subtracted from a larger one, what is the sign of the result ?

15. If any number be subtracted from a smaller one, what is the sign of the result? .

16. What is the number whose negative is the same as its positive?

For further exercises on this chapter, see Appendix, pp. 291293.

CHAPTER III

ADDITION AND SUBTRACTION

28. Definitions. A term of an expression is one of its elementary parts; that is, a part separated from other parts by the signs + or -. For example, the terms of the expression 3 x+2 are 3 x and 2; the terms of 5 ab - 6 are 5 ab and 6.

Like Terms are those that contain a common factor. Thus, 3 x, 4 x, and – 5 x are like terms because they contain the common factor x. Likewise, 2 ab, 3 ab, 4 ab, and 11 ab are like terms because they contain the common factor ab.

29. Adding Like Terms. If we observe what was said in § 11 about adding like numbers, we see that like terms may always be added by merely adding their separate coefficients to obtain a new coefficient, and then multiplying that new coefficient by the common factor.

For example, suppose the like terms to be added are 2 x, 3 x, -5x, –4 x, and 9 x. In this case, if we add the separate coefficients, we have 2+3+(-5)+(-4)+9, which reduces (by $ 17) to 6. Hence the answer is 6 x.

As another example, let us find the sum of the like terms 2 r2, -3 r2, 2 r2, – 7 r2, and 4 r2. The work may be arranged as follows: Adding the separate coefficients gives

2+(-3)+2+(-7)+4=-2. Therefore, the sum of the given terms is – 2 r2. Ans. Thus, we have the following rule.

RULE FOR ADDING LIKE TERMS. Add the coefficients for new coefficient and multiply it by the common factor.

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