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(2) By removing the parentheses and simplifying the result thus obtained. Thus,

10+(6+2)=10+6+2=18. Other illustrations, which should be carefully examined, follow. 2+(4-1)=2+3=5,

I 3+(5-7)=3+(-2)=1, or, 2+(4-1)=2+4–1=6–1=5. | or, 3+(5-7)=3+5–7 =8–7=1.

ORAL EXERCISES Work mentally each of the following exercises, first with parentheses and then without. 1. 10+(7+3).

7. (3-1)+(4-3). 2. 8+(9–6).

8. –4+(3-1). 3. (6+2)+4.

9. (1-2)+(1-3). 4. (10—6) +2.

10. 2+(4-3+1). 5. 1+(2-3).

11. (3-4-1)-1. 6. (2+3)+(3-2).

12. -1+3–2+(6—7). In each of the examples thus far considered the parentheses have been preceded by the sign +. Suppose now a case where the parentheses are preceded by a – sign. For example, consider the expression 10–(6+2). Here again we may proceed in either of two ways:

(1) By getting the value of the part in parentheses and subtracting it from 10. Thus,

10—(6+2)=10-8=2. (2) By simply removing the parentheses, provided, however, that we first change the sign of each term in the parentheses. (See $ 36.) Thus,

10—(6+2)=10–6–2=4–2=2. Other illustrations follow : 4-(3-2)=4-1=3,

I 2—(3–4)=2–(-1)=2+1=3, or, 4-(3-2)=4-3+2=1+2=3. or, 2— (3–4)=2–3+4=-1+4=3. ORAL EXERCISES Work mentally each of the following, first with parentheses, then without. 1. 10—(7+3).

6. 1+(2-1)-(3-1). 2. 8-(9–6).

7. –(3-2) – (2-3). 3. 7—(1+2-4).

8. –(2+3,1)+(2-1). 4. (2+3)-(4-5).

9. 2-4-(7–5). 5. -4-(3-1).

10.3+(2-4+1)-(3-7). We may now state what we have just seen in the following rule.

RULE FOR REMOVING PARENTHESES. A parenthesis preceded by the + sign (either expressed or understood) may always be removed.

A parenthesis preceded by the sign may be removed provided the sign of each term in the parenthesis be first changed.

EXERCISES

In each of the following exercises, remove the parentheses and reduce to the simplest form.

1. a+(6—c)-(2 a+3 b).

SOLUTION. Following the Rule of $ 40, the result of removing the parentheses is a+b-c-2 a-3 b. Combining like terms, this becomes – a-2b-c. Ans.

2. 2 a-(6-c)+(3 c-d). 3. (2x2 - y2) - (x+yz). 4. –(m2- n2)+(m-na+pq). 5. a+2 6+3 c-(a+b+c) – (2 a+3 b2 c). 6. a2b+b+c+ac (2 ab2 3 aạc)+(4 a2b 5 ac2 — 6 a262).

41. Bracket. Brace. Vinculum. When a group of terms is included within another group, it becomes necessary to use some other form than parentheses. The bracket [ ], the brace { }, and the vinculum are used for such purposes. For example, the expression

a+[r- {6+(6+c-d)}] means that the group (b+c-d) is first to be added to 6, then the result (considered as a new group) is to be subtracted from r, then this result (considered as a new group) is finally to be added to a.

42. Removing Group Signs. When an expression contains various group signs, such as the parenthesis, the bracket, brace, etc., they may all be removed in succession, beginning either with the outermost or innermost, preferably the latter.

Example. Simplify 8a-[3 6+4 a +(-a+26)].
SOLUTION. The rule of g 40 gives

8 a- [3 6+4 a+(-a+2 b)]

= 8 a— [3 6+4 a-a+2 b), which, combining terms,

=8.[3 6+3 a+2 b), which, by the rule of $ 40,

=8 a-3 6–3 a2 b, or, by combining terms,

=5 a-5 b. Ans.

WRITTEN EXERCISES Simplify each of the following expressions by removing all the signs of grouping. 1. a-(2a+4 a) – (5a+10). 3. 10r-(4r-{-3r–2}). 2. 6a+(5 a— [2 a+1]). 4. (2 r-c) – (5 r–2c).

16. c+b+d.

5. x_{x-(2 – 3 x)}. 6. 8 a-(-3 a+4)+(-2 a+10). 7. 6r-{10—(2r+6)-r}. 8. x-(10 x— {2 x+4}-6). 9. 20 z-{(2z+7 r) – (3 z+5 r)}. 10. 2(8-10 c) –[(-3+10 c)+(2-8 c)]. 11. 8 a-{4 a+[6 a, (2 a+17)]}. 12. {x— (x+(x-1]+4)+2}.

Find the value of each of the following expressions when a=4, b=3, c=2, and d=5. Recall the directions stated in § 15 for order of operations. 13. a-2(c+d). 14. abcd+Q2-C2. 16. 10 c2 – (3 c+2 d). 17. 3(a+b+c-d) — 6 a. 18. Va+c+d7-Vb+c+d-1. 19. (a+b)(ab).

20. (a+b)+8(dc).

22. 2 a+3 6 ** a-ab+bc

C-d+1° Solve each of the following equations. First remove parentheses in each. 23. 3 s-(s-10)=40. 24. (3 r–2)+(7 r-6)=10+(2r+4). 25. (4 x–5) - (2 x+7)=18-(x-1). 26. 2 a-(4 a+7)=(-a+2) – (2+5 a) 27. (m-6)+(m+6)=m+3. 28. [2 x—(2-x)]=3-(-1).

21. a?+62+02

EXERCISES — REVIEW OF CHAPTER IV

I. ORAL EXERCISES Read each of the following expressions. 1. 5(x−y+z).

6. (m,n)(m2-n2)(m3— n3). 2. (a+b)(a-6).

7. (Vx-1)(Vy+1+1). 3. (a +62)(a? — 62).

8. V(x-1)(y2+4). 4. (a+b)2(ab).

9. (25 – x4) V5(y2 +3). 5. 6(xy)(p-q+r). 10. 23—(3— 2 x2)2. 11. State the value of the expression in Ex. 7 when x=1, y=3.

12. Do a2 +62 and (a+b)2 mean the same thing? Are they equal? Explain.

13. Are a+b-c and -(c-a-b) equal to each other? Explain.

14. From among the following expressions pick out those which are equal to a+b-c:

(a) (a+b)-c. (d) a+(6-c). (g) a-(cb). (b) (ac)+b. (e) b-(a+c). (h) -C-(ab). (c) -c-(6-a). (f) -c+(a+b). (i) (6c)-(-a).

15. State such ways as you can for writing x-y+z, using parentheses.

II. WRITTEN EXERCISES When a=1, b=2, c=3, d=4, and e=5 find the value of each of the following expressions.

16. a-(e+b)–(c+d)-(e-d+b+c). 17. 3 ab? 2 be3 (dea dc2)+8 be. 18. V2 edb+4 e- 9 acc4 — 2 cod(abe-abcde). 19. Solve the equation 2 x-(4+x) – 5 x+20=4 x+(4-5x).

20. Solve the equation 10 x—3—(4-2 x)+(3 x-4x+5—2 x) =2–3x+4 x—(2 x+x)+7.

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