5.3
Adding and Subtracting Unlike
Fractions
5.3
OBJECTIVES
1. Write the sum of two unlike fractions in
simplest form
2. 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 1
Finding the LCD
Add
5
1
.
9
6
Step 1 To find the LCD, factor each denominator.
933
3 appears twice.
623
To form the LCD, include each factor the greatest number of times it appears in any single
denominator. In this example, use one 2, because 2 appears only once in the factorization
of 6. Use two 3s, because 3 appears twice in the factorization of 9. Thus the LCD for the
fractions in 2 3 3 18.
Step 2 “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?
© 2001 McGraw-Hill Companies
P
PR
Q
QR
5
52
10
9
92
18
1
13
3
6
63
18
Step 3 Add the fractions.
5
1
10
3
13
9
6
18
18
18
13
is in simplest form, and so we are done!
18
411
412
CHAPTER 5
ALGEBRAIC FRACTIONS
CHECK YOURSELF 1
Add.
(a)
1
3
6
8
(b)
3
4
10
15
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:
Step 1
Step 2
Step 3
Step 4
To Add or Subtract Unlike Fractions
Find the lowest common denominator of all the fractions.
Convert each fraction to an equivalent fraction with the LCD as a
denominator.
Add or subtract the like fractions formed in step 2.
Write the sum or difference in simplest form.
Example 2
Adding Unlike Fractions
(a) Add
3
4
2.
2x
x
Step 1 Factor the denominators.
2x 2 x
x2 x x
the denominators will be a
common denominator, it is not
necessarily the lowest common
denominator (LCD).
The LCD must contain the factors 2 and x. The factor x must appear twice because it
appears twice as a factor in the second denominator.
The LCD is 2 x x, or 2x2.
Step 2
3
3x
3x
2
2x
2x x
2x
4
42
8
2
2
x2
x 2
2x
Step 3
3
4
3x
8
3x 8
2 2 2
2x
x
2x
2x
2x2
The sum is in simplest form.
© 2001 McGraw-Hill Companies
NOTE Although the product of
ADDING AND SUBTRACTING UNLIKE FRACTIONS
(b) Subtract
SECTION 5.3
413
4
3
.
2 3x
2x3
Step 1 Factor the denominators.
3x2 3 x x
2x3 2 x x x
The LCD must contain the factors 2, 3, and x. The LCD is
2 3 x x x or 6x3
The factor x must appear
3 times. Do you see why?
Step 2
NOTE Both the numerator and
the denominator must be
multiplied by the same
quantity.
4
4 2x
8x
2
3
3x2
3x 2x
6x
3
33
9
3
3
2x3
2x 3
6x
Step 3
3
8x
9
8x 9
4
3 3 3
3x2
2x
6x
6x
6x3
The difference is in simplest form.
CHECK YOURSELF 2
Add or subtract as indicated.
(a)
5
3
2 3
x
x
(b)
1
3
2
5x
4x
We can also add fractions with more than one variable in the denominator. Example 3
shows this property.
Example 3
Adding Unlike Fractions
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Add
2
3
3.
3x2y
4x
Step 1 Factor the denominators.
3x2y 3 x x y
4x3 2 2 x x x
The LCD is 12x3y. Do you see why?
414
CHAPTER 5
ALGEBRAIC FRACTIONS
Step 2
2
2 4x
8x
2
3x2y
3x y 4x
12x3y
3
3 3y
9y
3
4x3
4x 3y
12x3y
Step 3
2
3
8x
9y
2 3 3 3x y
4x
12x y
12x3y
8x 9y
12x3y
CHECK YOURSELF 3
Add.
2
1
2 3x y
6xy2
Fractions with binomials in the denominator can also be added by taking the approach
shown in Example 3. Example 4 illustrates this approach with binomials.
Example 4
Adding Unlike Fractions
(a) Add
5
2
.
x
x1
Step 1 The LCD must have factors of x and x 1. The LCD is x(x 1).
Step 2
5
5(x 1)
x
x(x 1)
2
2x
x1
x(x 1)
Step 3
5
2
5(x 1)
2x
x
x1
x(x 1)
x(x 1)
5x 5 2x
x(x 1)
7x 5
x(x 1)
© 2001 McGraw-Hill Companies
NOTE The y in the numerator
and that in the denominator
cannot be divided out because
they are not factors.
ADDING AND SUBTRACTING UNLIKE FRACTIONS
(b) Subtract
SECTION 5.3
415
3
4
.
x2
x2
Step 1 The LCD must have factors of x 2 and x 2. The LCD is (x 2)(x 2).
Step 2
NOTE Multiply numerator and
denominator by x 2.
NOTE Multiply numerator and
denominator by x 2.
3
3(x 2)
x2
(x 2)(x 2)
4
4(x 2)
x2
(x 2)(x 2)
Step 3
4
3(x 2) 4(x 2)
3
x2
x2
(x 2)(x 2)
Note that the x term becomes negative and the
constant term becomes positive.
3x 6 4x 8
(x 2)(x 2)
x 14
(x 2)(x 2)
CHECK YOURSELF 4
Add or subtract as indicated.
(a)
3
5
x2
x
(b)
2
4
x3
x3
Example 5 will show how factoring must sometimes be used in forming the LCD.
Example 5
© 2001 McGraw-Hill Companies
Adding Unlike Fractions
(a) Add
3
5
.
2x 2
3x 3
Step 1 Factor the denominators.
2x 2 2(x 1)
C A U TI O N
x 1 is not used twice in
forming the LCD.
3x 3 3(x 1)
The LCD must have factors of 2, 3, and x 1. The LCD is 2 3(x 1), or 6(x 1).
416
CHAPTER 5
ALGEBRAIC FRACTIONS
Step 2
3
3
33
9
2x 2
2(x 1)
2(x 1) 3
6(x 1)
5
5
52
10
3x 3
3(x 1)
3(x 1) 2
6(x 1)
Step 3
3
5
9
10
2x 2
3x 3
6(x 1)
6(x 1)
(b) Subtract
9 10
6(x 1)
19
6(x 1)
3
6
2
.
2x 4
x 4
Step 1 Factor the denominators.
2x 4 2(x 2)
x2 4 (x 2)(x 2)
The LCD must have factors of 2, x 2, and x 2. The LCD is 2(x 2)(x 2).
Step 2
NOTE Multiply numerator and
denominator by x 2.
NOTE Multiply numerator and
denominator by 2.
3
3
3(x 2)
2x 4
2(x 2)
2(x 2)(x 2)
6
6
62
12
x2 4
(x 2)(x 2)
2(x 2)(x 2)
2(x 2)(x 2)
Step 3
NOTE Remove the parentheses
and combine like terms in the
numerator.
3x 6 12
2(x 2)(x 2)
3x 6
2(x 2)(x 2)
Step 4 Simplify the difference.
1
NOTE Factor the numerator
and divide by the common
factor x 2.
3(x 2)
3
3x 6
2(x 2)(x 2)
2(x 2)(x 2)
2(x 2)
1
© 2001 McGraw-Hill Companies
3
6
3(x 2) 12
2
2x 4
x 4
2(x 2)(x 2)
ADDING AND SUBTRACTING UNLIKE FRACTIONS
(c) Subtract
SECTION 5.3
5
2
2
.
x 1
x 2x 1
2
Step 1 Factor the denominators.
x2 1 (x 1)(x 1)
x2 2x 1 (x 1)(x 1)
The LCD is (x 1)(x 1)(x 1).
Two factors are needed.
Step 2
5(x 1)
5
(x 1)(x 1)
(x 1)(x 1)(x 1)
2(x 1)
2
(x 1)(x 1)
(x 1)(x 1)(x 1)
Step 3
5
2
5(x 1) 2(x 1)
2
x2 1
x 2x 1
(x 1)(x 1)(x 1)
NOTE Remove the parentheses
and simplify in the numerator.
5x 5 2x 2
(x 1)(x 1)(x 1)
3x 7
(x 1)(x 1)(x 1)
CHECK YOURSELF 5
Add or subtract as indicated.
(a)
5
1
2x 2
5x 5
(c)
4
3
2
x x2
x 4x 3
(b)
3
1
x2 9
2x 6
2
Recall from Section 5.1 that
© 2001 McGraw-Hill Companies
a b (b a)
Let’s see how this can be used in adding or subtracting algebraic fractions.
Example 6
Adding Unlike Fractions
Add
4
2
.
x5
5x
417
418
CHAPTER 5
ALGEBRAIC FRACTIONS
Rather than try a denominator of (x 5)(5 x), let’s simplify first.
4
2
4
2
x5
5x
x5
(x 5)
a
a
b
b
4
2
x5
x5
The LCD is now x 5, and we can combine the fractions as
42
x5
2
x5
CHECK YOURSELF 6
Subtract.
3
1
x3
3x
CHECK YOURSELF ANSWERS
13
17
5x ; (b)
2. (a)
24
30
x3
8x 10
2x 18
4. (a)
; (b)
x(x 2)
(x 3)(x x 18
(c)
6.
(x 1)(x 2)(x 3)
1. (a)
3
; (b)
12x 5
20x2
3)
5. (a)
3.
4y x
6x2y2
27
1
; (b)
;
10(x 1)
2(x 3)
4
x3
© 2001 McGraw-Hill Companies
NOTE Replace 5 x with
(x 5). We now use the fact
that
Name
5.3
Exercises
Section
Date
Add or subtract as indicated. Express your result in simplest form.
1.
3.
5.
7.
3
5
7
6
13
7
25
20
2.
4.
y
3y
4
5
6.
7a
a
3
7
8.
7
4
12
9
3
7
5
9
5x
2x
6
3
3m
m
4
9
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
9.
3
4
x
5
5
a
11.
a
5
13.
5
3
2
m
m
10.
5
2
x
3
y
3
12.
3
y
14.
3
4
2 x
x
11.
12.
13.
14.
15.
16.
17.
15.
2
5
2 x
7x
7
5
17.
2
9s
s
19.
3
5
3
4b2
3b
16.
7
5
3
3w
w
11
5
18. 2 x
7x
20.
4
3
2
5x3
2x
18.
19.
20.
21.
© 2001 McGraw-Hill Companies
22.
21.
x
2
x2
5
y
3
23.
y4
4
25.
4
3
x
x1
22.
3
a
4
a1
m
2
24.
m3
3
26.
2
1
x
x2
23.
24.
25.
26.
419
ANSWERS
27.
27.
5
2
a1
a
28.
3
4
x2
x
29.
4
2
2x 3
3x
30.
7
3
2y 1
2y
31.
2
3
x1
x3
32.
5
2
x1
x2
33.
4
1
y2
y1
34.
5
3
x4
x1
35.
2
3
b3
2b 6
36.
4
3
a5
4a 20
37.
x
2
x4
3x 12
38.
x
5
x3
2x 6
39.
4
1
3m 3
2m 2
40.
3
2
5y 5
3y 3
41.
4
1
5x 10
3x 6
42.
2
5
3w 3
2w 2
43.
7
2c
3c 6
7c 14
44.
5
4c
3c 12
5c 20
45.
y1
y
y1
3y 3
46.
x2
x
x2
3x 6
47.
3
2
x2 4
x2
48.
4
3
2
x2
x x2
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
44.
45.
46.
47.
48.
420
© 2001 McGraw-Hill Companies
43.
ANSWERS
49.
3x
1
x 3x 2
x2
50.
a
4
a 1
a1
51.
2x
4
x 5x 6
x2
52.
7a
4
a a 12
a4
2
2
2
49.
50.
2
2
1
53.
3x 3
4x 4
2
3
54.
5w 10
2w 4
4
3
55.
3a 9
2a 4
2
3
56.
3b 6
4b 8
51.
53.
54.
57.
5
3
2
x 16
x x 12
58.
3
1
2
x 4x 3
x 9
59.
2
3y
2
y y6
y 2y 15
60.
3
2a
2
a a 12
a 2a 8
56.
6x
5x
2
x 9
x x6
62.
4y
2y
2
y 6y 5
y 1
57.
61.
2
2
2
2
52.
55.
2
2
58.
3
2
63.
a7
7a
65.
5
3
64.
x5
5x
2x
1
2x 3
3 2x
66.
9m
3
3m 1
1 3m
Add or subtract, as indicated.
67.
60.
61.
1
1
2a
2
a3
a3
a 9
68.
1
1
4
2
p1
p3
p 2p 3
2x2 3x
7x
x2 3x 21
2
2
69. 2
x 2x 63
x 2x 63
x 2x 63
70. 59.
3 2x2
4x2 2x 1
2x2 3x
x2 9x 20
x2 9x 20
x2 9x 20
71. Consecutive integers. Use a rational expression to represent the sum of the
62.
63.
64.
65.
66.
67.
68.
69.
70.
reciprocals of two consecutive even integers.
71.
© 2001 McGraw-Hill Companies
72. Integers. One number is two less than another. Use a rational expression to represent
the sum of the reciprocals of the two numbers.
73. Refer to the rectangle in the figure. Find an expression that represents its perimeter.
72.
73.
2x 1
5
4
3x 1
421
ANSWERS
74.
74. Refer to the triangle in the figure. Find an expression that represents its perimeter.
a.
3
1
x2
4x
b.
5
x2
c.
d.
Getting Ready for Section 5.4 [Section 0.2]
e.
Perform the indicated operations.
2 4
3 5
4
8
(c) 7
5
5 16
(e) 8 15
15 24
(g)
8 25
f.
(a)
g.
h.
5 4
6 11
1
7
(d) 6
9
15
10
(f)
21
7
28
21
(h)
16
20
(b)
Answers
53
17
17y
46a
15 4x
25 a2
3.
5.
7.
9.
11.
42
100
20
21
5x
5a
5m 3
14 5x
7s 45
9b 20
7x 4
13.
15.
17.
19.
21.
m2
7x2
9s2
12b3
5(x 2)
y 12
7x 4
3a 2
2(8x 3)
23.
25.
27.
29.
4(y 4)
x(x 1)
a(a 1)
3x(2x 3)
5x 9
3(y 2)
7
3x 2
31.
33.
35.
37.
(x 1)(x 3)
(y 2)(y 1)
2(b 3)
3(x 4)
11
7
49 6c
2y 3
39.
41.
43.
45.
6(m 1)
15(x 2)
21(c 2)
3(y 1)
2x 1
2x 1
6
5x 11
47.
49.
51.
53.
(x 2)(x 2)
(x 1)(x 2)
x3
12(x 1)(x 1)
a 43
2x 3
3y2 4y 10
55.
57.
59.
6(a 3)(a 2)
(x 4)(x 4)(x 3)
(y 3)(y 2)(y 5)
x
1
2x 1
2
x2
61.
63.
65.
67.
69.
(x 3)(x 2)
a7
2x 3
a3
x9
2x 2
2(6x2 5x 21)
8
10
5
3
71.
73.
a.
b.
c.
d.
x(x 2)
5(3x 1)
15
33
14
14
2
1
9
5
e.
f.
g.
h.
3
2
5
3
422
© 2001 McGraw-Hill Companies
1.