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. rstry—bs—by cs+cy-ks – ky 3. 4x2–9 M 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 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 13. Ōi a-bTa-)2 4. 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(a−b) – 3(6–a)' Perform the multiplication indicated in each of the following exercises. 21. (x-1)(2+%). 22. (ab + ab) (ab – ab.). 25. 14—54. r-s. p2 .. (r—8)2 72+rs p2+52 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. a4–64 a2 +62 1 /2 a+ 2 a+b_u). a2—2 ab+62 ' a2 - ab lato Tato 7 X2 – 13 x+22 . x2–5x+6 u. 22–9 x+8 * x2 – 6 x – 16 32. (a+4)=(a-). 36. (r+s) = (47+1). 39. 28 (2 a(a? — m2). a?—ar). a2+2 ar+r2. *40. 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, + 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. x=7. Ans. 3 6 _5_4 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. or ORAL EXERCISES Solve each of the following equations by clearing of fractions. 11. -2=0. 1. Ž=1. 2. =1. 3. 1 6. -2=1. 12. 2-1=0. 13. 4+,-0. |