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. 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: Perform the multiplication indicated in each of the following exercises. 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. 2, a 2, x+y, a+b. 3' 2' a 2 a-b 28. State the reciprocal of 5, 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. 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 = 12, an equation cleared of fractions. x= x 1 3 12 Multiplying both sides of the equation + 4x+1=3, an equation cleared of fractions. 1 by 12, we get 4 EXAMPLE 1. Solve the fractional equation 3 4 6 SOLUTION. Multiplying both sides of the equation by 12, the L. C. M. of the denominators, we find, 8x+9x+10x=108. Combining like terms, 27 x=108. Dividing by 27, x=4. Ans. CHECK. Substituting 4 for x in the given equation, we have SOLUTION. Multiplying both sides by 8, the L. C. M. of the 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. ORAL EXERCISES Solve each of the following equations by clearing of fractions. |