Adding and Subtracting Unlike Fractions

Transcription

1 . Adding and Subtracting Unlike Fractions. OBJECTIVES. Write the sum of two unlike fractions in simplest form. Write the difference of two unlike fractions in simplest form Adding or subtracting unlike fractions (fractions that do not have the same denominator) requires a bit more work than adding or subtracting the like fractions of the previous section. When the denominators are not the same, we must use the idea of the lowest common denominator (LCD). Each fraction is built up to an equivalent fraction having the LCD as a denominator. You can then add or subtract as before. Let s review with an example from arithmetic. Example Finding the LCD Add. 9 6 Step To find the LCD, factor each denominator. 9 appears twice. 6 To form the LCD, include each factor the greatest number of times it appears in any single denominator. In this example, use one, because appears only once in the factorization of 6. Use two s, because appears twice in the factorization of 9. Thus the LCD for the fractions in 8. Step Build up each fraction to an equivalent fraction with the LCD as the denominator. Do this by multiplying the numerator and denominator of the given fractions by the same number. NOTE Do you see that this uses the fundamental principle in the following form? P PR Q QR McGraw-Hill Companies Step 8 Add the fractions is in simplest form, and so we are done!

2 CHAPTER ALGEBRAIC FRACTIONS CHECK YOURSELF Add. (a) 6 8 (b) 0 The process of finding the sum or difference is exactly the same in algebra as it is in arithmetic. We can summarize the steps with the following rule: Step by Step: To Add or Subtract Unlike Fractions Step Step Find the lowest common denominator of all the fractions. Convert each fraction to an equivalent fraction with the LCD as a denominator. Step Add or subtract the like fractions formed in step. Step Write the sum or difference in simplest form. Example Adding Unlike Fractions (a) Add. x x NOTE Although the product of the denominators will be a common denominator, it is not necessarily the lowest common denominator (LCD). Step Factor the denominators. x x x x x The LCD must contain the factors and x. The factor x must appear twice because it appears twice as a factor in the second denominator. The LCD is x x, or x. Step x x x x x x x x 8 x Step x x x x 8 x 8 x x The sum is in simplest form. 00 McGraw-Hill Companies

3 ADDING AND SUBTRACTING UNLIKE FRACTIONS SECTION. (b) Subtract. x x Step Factor the denominators. x x x x x x x The LCD must contain the factors,, and x. The LCD is NOTE Both the numerator and the denominator must be multiplied by the same quantity. x x x or 6x Step x x 8x x x 6x x x 9 6x The factor x must appear times. Do you see why? Step x 8x x 6x 9 8x 9 6x 6x The difference is in simplest form. CHECK YOURSELF Add or subtract as indicated. (a) x x (b) x x We can also add fractions with more than one variable in the denominator. Example shows this property. Example Adding Unlike Fractions 00 McGraw-Hill Companies Add. x y x Step Factor the denominators. x y x x y x x x x The LCD is x y. Do you see why?

4 CHAPTER ALGEBRAIC FRACTIONS Step x y x x x y x y x y 8x x y 9y x y Step NOTE The y in the numerator and that in the denominator cannot be divided out because they are not factors. x y x 8x x y 8x 9y x y 9y x y CHECK YOURSELF Add. x y 6xy Fractions with binomials in the denominator can also be added by taking the approach shown in Example. Example illustrates this approach with binomials. Example Adding Unlike Fractions (a) Add. x x Step The LCD must have factors of x and x. The LCD is x(x ). Step (x ) x x(x ) x x x(x ) Step x (x ) x x(x ) x x x(x ) x x(x ) x x(x ) 00 McGraw-Hill Companies

5 ADDING AND SUBTRACTING UNLIKE FRACTIONS SECTION. (b) Subtract. x x Step The LCD must have factors of x and x. The LCD is (x )(x ). Step NOTE Multiply numerator and denominator by x. NOTE Multiply numerator and denominator by x. x (x ) (x )(x ) x (x ) (x )(x ) Step x (x ) (x ) x (x )(x ) Note that the x term becomes negative and the constant term becomes positive. x 6 x 8 (x )(x ) x (x )(x ) CHECK YOURSELF Add or subtract as indicated. (a) x x (b) x x Example will show how factoring must sometimes be used in forming the LCD. Example Adding Unlike Fractions 00 McGraw-Hill Companies CAUTION x is not used twice in forming the LCD. (a) Add. x x Step Factor the denominators. x (x ) x (x ) The LCD must have factors of,, and x. The LCD is (x ), or 6(x ).

6 6 CHAPTER ALGEBRAIC FRACTIONS Step x (x ) (x ) 9 6(x ) x (x ) (x ) 0 6(x ) Step x x 9 6(x ) 0 6(x ) (b) Subtract 9 0 6(x ) 9 6(x ) x 6 x. Step Factor the denominators. x (x ) x (x )(x ) The LCD must have factors of, x, and x. The LCD is (x )(x ). Step NOTE Multiply numerator and denominator by x. NOTE Multiply numerator and denominator by. x (x ) (x ) (x )(x ) 6 x 6 (x )(x ) 6 (x )(x ) (x )(x ) Step x 6 (x ) x (x )(x ) NOTE Remove the parentheses and combine like terms in the numerator. NOTE Factor the numerator and divide by the common factor x. Step x 6 (x )(x ) x 6 (x )(x ) Simplify the difference. x 6 (x )(x ) (x ) (x )(x ) (x ) 00 McGraw-Hill Companies

7 ADDING AND SUBTRACTING UNLIKE FRACTIONS SECTION. (c) Subtract x x x. Step Factor the denominators. x (x )(x ) x x (x )(x ) The LCD is (x )(x )(x ). Step Step Two factors are needed. (x )(x ) (x ) (x )(x )(x ) (x )(x ) (x ) (x )(x )(x ) x (x ) (x ) x x (x )(x )(x ) NOTE Remove the parentheses and simplify in the numerator. x x (x )(x )(x ) x (x )(x )(x ) CHECK YOURSELF Add or subtract as indicated. (a) (c) x x x x x x (b) x 9 x 6 00 McGraw-Hill Companies Recall from Section. that a b (b a) Let s see how this can be used in adding or subtracting algebraic fractions. Example 6 Adding Unlike Fractions Add. x x

8 8 CHAPTER ALGEBRAIC FRACTIONS NOTE Replace x with (x ). We now use the fact that a a b b Rather than try a denominator of (x )( x), let s simplify first. x x x (x ) The LCD is now x, and we can combine the fractions as x x x x CHECK YOURSELF 6 Subtract. x x CHECK YOURSELF ANSWERS x x y x. (a) ; (b). (a) ; (b). 0 x 0x 6x y 8x 0 x 8. (a) ; (b). (a) ; (b) ; x(x ) (x )(x ) 0(x ) (x ) x 8 (c) 6. (x )(x )(x ) x 00 McGraw-Hill Companies

9 Name. Exercises Section Date Add or subtract as indicated. Express your result in simplest form ANSWERS.... y y. 6. x 6 x a. 8. a m m x x.... a a y y..... m m. 6. x x x x w w s s x x b b x x. 00 McGraw-Hill Companies x.. x y.. y. 6. x x a a m m x x

10 ANSWERS.. 8. a a x x x x y y.... x x x x.... y y x x b b 6 a a x. 8. x x x x x m m y y.... x 0 x 6 w w c 6 c c y. 6. y y y. 8. x x c x x c c 0 x x 6 x x x 00 McGraw-Hill Companies 0

11 ANSWERS x x x x x.. x x 6 x.. x x. 6. a 9 a. 8. x 6 x x y y 6 y y y 6x x 9 x x x a a x x x Add or subtract, as indicated. a a a a a a a w 0 w b 6 b 8 x x x 9 a a a a a 8 y y 6y x x y y 9m m m a a a a x x x x 6 x x x 6 x x x x 6 p p p p x x 9x 0 x x x 9x 0 x x x 9x McGraw-Hill Companies. Consecutive integers. Use a rational expression to represent the sum of the reciprocals of two consecutive even integers.. Integers. One number is two less than another. Use a rational expression to represent the sum of the reciprocals of the two numbers.. Refer to the rectangle in the figure. Find an expression that represents its perimeter. x x

12 ANSWERS.. Refer to the triangle in the figure. Find an expression that represents its perimeter. a. x x b. x c. d. Getting Ready for Section. [Section 0.] e. f. g. h. Perform the indicated operations. (a) (c) 8 (e) 6 8 (g) 8 (b) (d) (f) (h) Answers. y 6a x a x a. m x s 9b 0 x m x 9s b (x ). y x a (8x ).. 9. (y ) x(x ) a(a ) x(x ). x 9 (y ) x... (x )(x ) (y )(y ) (b ) (x ) c y... 6(m ) (x ) (c ) (y ). x x 6 x 9... (x )(x ) (x )(x ) x (x )(x ). a x y y (a )(a ) (x )(x )(x ) (y )(y )(y ) 6. x x x (x )(x ) a x a x 9. x (6x x ) 8 0. a. b. c. d. x(x ) (x ) e. 9 f. g. h. 00 McGraw-Hill Companies