§ 6.2 Adding and Subtracting Rational Expressions.
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§ 6.2
Adding and Subtracting Rational Expressions
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 6.2
Adding Rational Expressions
In this section, you will practice adding and subtracting rational expressions. Rememberthat when adding or subtracting fractions, it is necessary to rewrite the fractionsas fractions having the same denominator, which is called the common denominator for the fractions being combined.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.2
Adding Rational Expressions
Adding Rational Expressions With Common Denominators
If are rational expressions, then
To add rational expressions with the same denominator, add numerators and place the sum over the common denominator. If possible, simplify the result.
R
Q
R
P and
R
QP
R
Q
R
P
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 6.2
Adding Rational Expressions
EXAMPLEEXAMPLE
Add: .33
22
2
2
2
xx
xx
xx
xx
SOLUTIONSOLUTION
Add numerators. Place this sum over the common denominator.
Factor.
This is the original expression.xx
xx
xx
xx
33
22
2
2
2
xx
xxxx
3
22
22
xx
xx
3
22
2
3
12
xx
xx
Combine like terms.
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 6.2
Adding Rational Expressions
Simplify.
Factor and simplify by dividing out the common factor, x.
3
12
xx
xx
CONTINUECONTINUEDD
3
12
x
x
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 6.2
Adding Rational Expressions
Check Point 1Check Point 1
Add:
SOLUTIONSOLUTION
Add numerators. Place this sum over the common denominator.
Factor.
Combine like terms.
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 6.2
Subtracting Rational Expressions
Subtracting Rational Expressions With Common Denominators
If are rational expressions, then
To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, simplify the result.
R
Q
R
P and
R
QP
R
Q
R
P
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 6.2
Subtracting Rational Expressions
EXAMPLEEXAMPLE
Subtract: .6
12
6
2622
2
xx
x
xx
xx
SOLUTIONSOLUTION
Subtract numerators. Place this difference over the common denominator.
This is the original expression.
Remove the parentheses and distribute.
6
12
6
2622
2
xx
x
xx
xx
6
12262
2
xx
xxx
6
12262
2
xx
xxx
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 6.2
Subtracting Rational Expressions
Factor.
Combine like terms.
Factor and simplify by dividing out the common factor, x + 3.
6
342
2
xx
xx
CONTINUECONTINUEDD
23
31
xx
xx
23
31
xx
xx
2
1
x
xSimplify.
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 6.2
Adding Rational Expressions
Check Point 2Check Point 2
Subtract:
SOLUTIONSOLUTION
Subtract numerators. Place difference over the common denominator.
Factor.
Combine like terms.
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 6.2
Least Common Denominators
Finding the Least Common Denominator (LCD)
1) Factor each denominator completely.
2) List the factors of the first denominator.
3) Add to the list in step 2 any factors of the second denominator that do not appear in the list.
4) Form the product of each different factor from the list in step 3. This product is the least common denominator.
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 6.2
Least Common Denominators
EXAMPLEEXAMPLE
Find the LCD of: .472
and 20
322 yy
y
yy
SOLUTIONSOLUTION
1) Factor each denominator completely.
45202 yyyy
412472 2 yyyy
2) List the factors of the first denominator.
4 ,5 yy
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 6.2
Least Common Denominators
3) Add any unlisted factors from the second denominator. The second denominator is (2y - 1)(y + 4). One factor of y + 4 is already in our list, but the factor 2y – 1 is not. We add the factor 2y – 1 to our list.
4) The least common denominator is the product of all factors in the final list. Thus,
12 ,4 ,5 yyy
CONTINUECONTINUEDD
1245 yyy
is the least common denominator.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 6.2
Least Common Denominators
Check Point Check Point 33
Find the LCD of: .9
2 and
6
72 xx
SOLUTIONSOLUTION
1) Factor each denominator completely.
2) List the factors of the first denominator.
3) Add any unlisted factors from the second denominator.
4) The least common denominator is the product of all factors in the final list.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 6.2
Least Common Denominators
Check Point Check Point 44
Find the LCD of: .96
9 and
155
722 xxxx
SOLUTIONSOLUTION
1) Factor each denominator completely.
2) List the factors of the first denominator.
3) Add any unlisted factors from the second denominator.
4) The least common denominator is the product of all factors in the final list.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 6.2
Add & Subtract Fractions
Adding and Subtracting Rational Expressions That Have Different Denominators
1) Find the LCD of the rational expressions.
2) Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
3) Add or subtract numerators, placing the resulting expression over the LCD.
4) If possible, simplify the resulting rational expressions.
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 6.2
Adding Fractions
EXAMPLEEXAMPLE
Add: .23
3
82
722
xxxx
x
SOLUTIONSOLUTION
1) Find the least common denominator. Begin by factoring the denominators.
24822 xxxx
21232 xxxx
The factors of the first denominator are x + 4 and x – 2. The only factor from the second denominator that is unlisted is x – 1. Thus, the least common denominator is,
. 124 xxx
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 6.2
Adding Fractions
2) Write equivalent expressions with the LCD as denominators.
This is the original expression.
CONTINUECONTINUEDD
23
3
82
722
xxxx
x
21
3
24
7
xxxx
xFactored denominators.
421
43
124
17
xxx
x
xxx
xx Multiply each numerator and denominator by the extra factor required to form the LCD.
. 124 xxx
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 6.2
Adding Fractions
3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible.
Add numerators.
CONTINUECONTINUEDD
Perform the multiplications using the distributive property.
124
4317
xxx
xxx
124
12377 2
xxx
xxx
124
1247 2
xxx
xxCombine like terms.
Blitzer, Intermediate Algebra, 5e – Slide #20 Section 6.2
Adding Fractions
CONTINUECONTINUEDD
Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,
.124
1247 2
xxx
xx
Blitzer, Intermediate Algebra, 5e – Slide #21 Section 6.2
Adding Fractions
Check Point Check Point 55
Add
SOLUTIONSOLUTIONx
xx
33
32
Multiply
Like fractions
Simplified
233 xx
Blitzer, Intermediate Algebra, 5e – Slide #22 Section 6.2
Adding Fractions
Check Point Check Point 66
Add
SOLUTIONSOLUTION4
4
x
x
Multiply
Like fractions
Simplify
Combine like terms
Blitzer, Intermediate Algebra, 5e – Slide #23 Section 6.2
Subtracting Fractions
EXAMPLEEXAMPLE
Subtract: .65
3
67
1222
xx
x
xx
x
SOLUTIONSOLUTION
1) Find the least common denominator. Begin by factoring the denominators.
16672 xxxx
61652 xxxx
The factors of the first denominator are x - 4 and x – 1. The only factor from the second denominator that is unlisted is x + 1. Thus, the least common denominator is,
. 116 xxx
Blitzer, Intermediate Algebra, 5e – Slide #24 Section 6.2
Subtracting Fractions
2) Write equivalent expressions with the LCD as denominators.
This is the original expression.
CONTINUECONTINUEDD
Factored denominators.
Multiply each numerator and denominator by the extra factor required to form the LCD.
65
3
67
1222
xx
x
xx
x
61
3
16
12
xx
x
xx
x
161
13
116
112
xxx
xx
xxx
xx
. 116 xxx
Blitzer, Intermediate Algebra, 5e – Slide #25 Section 6.2
Subtracting Fractions
3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible.
Subtract numerators.
CONTINUECONTINUEDD
Perform the multiplications using the distributive property and FOIL.
Remove parentheses.
116
13112
xxx
xxxx
116
33122 22
xxx
xxxxxx
116
33122 22
xxx
xxxxxx
Blitzer, Intermediate Algebra, 5e – Slide #26 Section 6.2
Subtracting Fractions
Combine like terms in the numerator.
CONTINUECONTINUEDD
116
42
xxx
xx
Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,
.116
42
xxx
xx
Blitzer, Intermediate Algebra, 5e – Slide #27 Section 6.2
Subtracting Fractions
Check Point Check Point 77Subtract
SOLUTIONSOLUTION
13
23
xx
xx
Multiply
Like fractions
Combine like terms and Simplify
Subtract (add the opposite)
Blitzer, Intermediate Algebra, 5e – Slide #28 Section 6.2
Adding and Subtracting of Fractions
EXAMPLEEXAMPLE
Perform the indicated operations: .1
4
32
2
3 2
xxx
x
x
x
SOLUTIONSOLUTION
1) Find the least common denominator. Begin by factoring the denominators.
313 xx
31322 xxxx
The factors of the first denominator are 1 and x – 3. The only factor from the second denominator that is unlisted is x + 1. We have already listed all factors from the third denominator. Thus, the least common denominator is,
111 xx
Blitzer, Intermediate Algebra, 5e – Slide #29 Section 6.2
Adding and Subtracting of Fractions
. 13or 131 xxxx
CONTINUECONTINUEDD
2) Write equivalent expressions with the LCD as denominator.
1
4
32
2
3 2
xxx
x
x
x This is the original expression.
1
4
13
2
3
xxx
x
x
x Factor the second denominator.
31
34
13
2
13
1
xx
x
xx
x
xx
xx Multiply each numerator and denominator by the extra factor required to form the LCD.
Blitzer, Intermediate Algebra, 5e – Slide #30 Section 6.2
Adding and Subtracting of Fractions
CONTINUECONTINUEDD 3) & 4) Add and subtract numerators, putting this result
over the LCD. Simplify if possible.
13
3421
xx
xxxx Add and subtract numerators.
13
12422
xx
xxxx Perform the multiplications using the distributive property.
13
1422
xx
xxCombine like terms in the numerator.
Blitzer, Intermediate Algebra, 5e – Slide #31 Section 6.2
Adding and Subtracting of Fractions
CONTINUECONTINUEDD
.13
1422
xx
xx
Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,
Blitzer, Intermediate Algebra, 5e – Slide #32 Section 6.2
Addition of Fractions (opposites)
EXAMPLEEXAMPLE
Add: .109
22 xyyx
x
SOLUTIONSOLUTION
xyyx
x
109
22
xyyxyx
x
109
xyyxyx
x
10
1
19
xyyxyx
x
109
This is the original expression.
Factor the first denominator.
Multiply the numerator and the denominator of the second rational expression by -1.
Multiply by -1.
Blitzer, Intermediate Algebra, 5e – Slide #33 Section 6.2
Addition of Fractions (opposites)
yxyxyx
x
109
Rewrite –y + x as x – y.
Notice the LCD is (x + y)(x – y).
yxyx
yx
yxyx
x
109 Multiply the second numerator
and denominator by the extra factor required to form the LCD.
yxyx
yx
yxyx
x
10109 Perform the multiplications
using the distributive property.
CONTINUECONTINUEDD
Blitzer, Intermediate Algebra, 5e – Slide #34 Section 6.2
Addition of Fractions (opposites)
yxyx
yxx
10109
Add and subtract numerators.
yxyx
yxx
10109 Remove parentheses and
distribute.
yxyx
yx
1019 Combine like terms in the
numerator.
Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,
.1019
yxyx
yx
CONTINUECONTINUEDD
Blitzer, Intermediate Algebra, 5e – Slide #35 Section 6.2
Addition of Fractions
Important to Important to RememberRemember
Adding or Subtracting Rational Expressions
If the denominators are the same, add or subtract the numerators and place theresult over the common denominator.
If the denominators are different, write all rational expressions with the least common denominator (LCD). Once all rational expressions are written in terms of the LCD, then add or subtract as described above.
In either case, simplify the result, if possible. Even when you have used the LCD, it may be true that the sum of the fractions can be reduced.
Blitzer, Intermediate Algebra, 5e – Slide #36 Section 6.2
Addition of Fractions
Important to Important to RememberRememberFinding the Least Common Denominator (LCD)
The LCD is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.
That is - After factoring the denominators completely, the LCD can be determined by taking each factor to the highest power it appears in any factorization.
The Mathematics Teacher magazine accused the LCD of keeping up with the Joneses. The LCD wants everything (all of the factors) the other denominators have.
DONE
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