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EXERCISES — REVIEW OF CHAPTER IX 1. What principle of fractions is used in reducing fractions to lowest terms?

Reduce to lowest terms: 2 a—2 a-15

4 23 – 3 x2-4 x a2+2 a-35

23—8 x2 +16 x

5. rstrybsby
4 x2 – 12 x+9

cs+cy-ks ky
ala+26)4(a3+2 alb+aba)
0. 6(a2-4 62)2(a5 — 2 a3b2+ab4

3.

4x2–9

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Since an integer may be regarded as a fraction with the denominator 1 (see Note, $ 101) reduce each of the following expressions to a single fraction.

7. a+d?–ab.

8. 2-4-6-8.

a

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m2

9. 1+m+ m2

a . Perform the indicated additions and subtractions in the following exercises.

a-b a . 40. 4x+24' 6 x+36

atay a?- y2 12 2 _ 2 a _ 2 ab

10

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13. Ōi a-bTa-)2

4.
a-1'a+1 a? —2 a+1°
2 a+3 , a-4_3 a2—8 a-27
a+3 'a-5 Q2–2 a-15

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20. Changing the signs of the numerator and denominator of a fraction is the same as multiplying the numerator and denominator by –1. Apply this to explain each of the following reductions : 4 -4 . a

- a az 62 62-a23(ab) – 3(6–a)' Perform the multiplication indicated in each of the following exercises.

21. (x-1)(2+%).

22. (ab + ab) (ab – ab.).

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25. 14—54. r-s. p2 ..

(r—8)2 72+rs p2+52
26 x(xy) _._*(x+y)
x2+2 xy+y2 22 – 2 xy + y2
a-1 V

2 a2+6 a+5/ 12+7 a+12 The reciprocal of a number is 1 divided by that number. Thus, the reciprocal of x is 1/x; of ab is 1/ab; of a/b is 1/(a/b), or b/a. 28. State the reciprocal of 5, , , *T!,

< 2 a 2 x+y a+b.

'2' a' 2 'a-6 29. Show that multiplying a/b by the reciprocal of a gives the same result as dividing a/b by a.

30. Show that the sum of any two numbers, as a and b, divided by their product, is equal to the sum of their reciprocals.

Perform the operations indicated in each of the following exercises. a464 a2 +62

1 /2 a+
b

2 a+b_u). a22 ab+62 ' a2 - ab

lato

Tato 7 X2 – 13 x+22 . x2–5x+6 u. 22–9 x+8 * x2 – 6 x – 16

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32. (a+4)=(a-).
33. (6a+3) = 4,672.
37. (1*+246) + 471

36. (r+s) = (47+1).

39.

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28 (2 a(a? — m2). a?—ar). a2+2 ar+r2. *40.
30. 15 6(d2 — p2) bd+br/ 222 dr+p2 V.

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For further exercises on this chapter, see Appendix, pp. 301–303.

CHAPTER X

FRACTIONAL EQUATIONS

104. Fractional Equations. A fractional equation is one which contains fractional expressions.

105. Clearing of Fractions and Solving. Multiplying both members of an equation by such a number as will cancel each denominator is called clearing of fractions.

Thus, multiplying both sides of the equation x/2=6 by 2, we get x=12, an equation cleared of fractions. Multiplying both sides of the equation ;

2, we get 4 x+1=3, an equation cleared of fractions. EXAMPLE 1. Solve the fractional equation

2 x , 3 x 5 x –9.

3476 SOLUTION. Multiplying both sides of the equation by 12, the L. C. M. of the denominators, we find,

8x+9 x+10 x=108. Combining like terms,

27 x=108. Dividing by 27,

x=4. Ans. CHECK. Substituting 4 for x in the given equation, we have

84 12 + 20 =9,

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16+18+20=9,

which is a true statement.

0 3 EXAMPLE 2. Solve X-1_X-2

4 2 8 2 Solution. Multiplying both sides by 8, the L. C. M. of the denominators, we find 2 x–2–4 x+8=2–3–12. (Note the signs here

carefully.) Transposing,

-x+2 x-4 x=-3-12+2-8. Combining,

– 3 x=-21.
Dividing by -3,

x=7. Ans.
CHECK. Substituting 7 for x in the first equation gives
7-1_7-2 _ 7–3_

3 6 _5_4
4 2 8 2 0

or –1=-1, which is a true statement.

NOTE. In the given equation of Example 2 the sign before the fraction (x-2)/2 is minus. The line between the numerator and the denominator has the same effect as a parenthesis around the numerator, for when the equation is cleared of fractions and this line is removed the sign of each term in the numerator is changed. See § 93.

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ORAL EXERCISES Solve each of the following equations by clearing of fractions.

11. -2=0.

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1. Ž=1. 2. =1. 3. 1

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6. -2=1.
7.-4.
8. =3.

12. 2-1=0. 13. 4+,-0.

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