11-5
Adding and Subtracting Rational
Expressions
TEKS FOCUS
VOCABULARY
ĚComplex fraction – a rational expression that has at
TEKS (7)(F) Determine the sum, difference, product, and
quotient of rational expressions with integral exponents of
degree one and of degree two.
least one fraction in its numerator or denominator
or both
TEKS (1)(A) Apply mathematics to problems arising in
everyday life, society, and the workplace.
ĚApply – use knowledge or information for a specific
purpose, such as solving a problem
Additional TEKS (6)(H)
ESSENTIAL UNDERSTANDING
To operate with rational expressions, you can use much of what you know about
operating with fractions.
Problem 1
P
TEKS Process Standard (1)(D)
Finding the Least Common Multiple
What is the LCM of 12x2y(x2 + 2x + 1) and 18xy 3(x2 + 5x + 4)?
W
How do you
determine the
exponent of each
factor for the LCM?
Use the exponent from
the expression that
has that factor to the
greatest power.
Step 1
S
Find the prime factors of each expression.
12x2y(x2 + 2x + 1) = 22
18xy 3(x2 + 5x + 4) = 2
# 3x2y(x + 1)2
# 32xy3(x + 1)(x + 4)
S
Step
2 Write the product of the prime factors, each raised to the greatest
power that occurs in either expression.
22
The LCM is 22
# 32x2y3(x + 1)2(x + 4)
# 32x2y3(x + 1)2(x + 4), or 36x2y3(x + 1)2(x + 4).
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Problem 2
P
Finding the Least Common Denominator
What is the LCD of each pair of rational expressions? Find the sum and difference
of the rational expressions in simplest form. State any restrictions on the variables.
2x y
A 15y , 6z
# 5y
6z = 2 # 3z
15y = 3
2
# 3 # 5yz = 30yz
Find the prime factors of each denominator.
Write the LCM.
The least common denominator is the least common multiple, 30yz.
To add or subtract rational expressions, rewrite them to have the LCD as each
denominator.
y
5y
# # 2z2z + 2 # 3z # 5y
y
2x
2x
15y + 6z = 3 5y
Rewrite each expression.
5y 2
4xz
= 30yz + 30yz
Is this the answer?
Maybe. Remember that
with rational expressions,
you must check whether
the expression is in
simplest form.
=
Multiply.
4xz + 5y 2
30yz
Add the numerators.
y
5y
# # 2z2z - 2 # 3z # 5y
y
2x
2x
15y - 6z = 3 5y
Rewrite each expression.
5y 2
4xz
= 30yz - 30yz
=
Multiply.
4xz - 5y 2
30yz
Subtract the numerators.
In this case, the numerators 4xz + 5y 2 and 4xz - 5y 2 have no factors other than 1,
so each expression is in simplest form. The sum of the pair of rational expressions
4xz + 5y 2
is 30yz
B
and the difference is
4xz - 5y 2
30yz
for y ≠ 0 and z ≠ 0.
16
2
,
x2 x2 − 9
The denominator x2 is already factored, and x2 - 9 = (x - 3)(x + 3). Since there
are no common factors, the LCD is x2(x - 3)(x + 3).
16
2
2
+
=
x2 x2 - 9 x2
484
3)(x + 3)
16
# (x(x -- 3)(x
#x
+
+ 3) (x - 3)(x + 3) x
=
2x 2 - 18
16x 2
+
x 2(x - 3)(x + 3) x 2(x - 3)(x + 3)
=
18x 2 - 18
x 2(x - 3)(x + 3)
=
18(x - 1)(x + 1)
x 2(x - 3)(x + 3)
Lesson 11-5 Adding and Subtracting Rational Expressions
2
2
continued on next page ▶
Problem 2
continued
The expression is already in simplest form because there are no common factors.
2 - 18
The sum of the pair of expressions is 18x
4
2 for x ≠ 0, x ≠ 3, and x ≠ -3.
x - 9x
16
2
2
=
x2 x2 - 9 x2
+ 3)
16
# (x(x -- 3)(x
#x
3)(x + 3) (x - 3)(x + 3) x
=
2x 2 - 18
16x 2
x 2(x - 3)(x + 3) x 2(x - 3)(x + 3)
=
- 14x 2 - 18
x 2(x - 3)(x + 3)
2
2
2 - 18
The difference of the expressions is - 14x
4
2 for x ≠ 0, x ≠ 3, and x ≠ -3.
x - 9x
Problem
bl
3
Adding Rational Expressions
What is the sum of the two rational expressions in simplest form?
State any restrictions on the variable. x + 2 2x − 1
x−1
How does the LCD
help you simplify this
sum?
The LCD is (x - 1)(x - 2).
Multiply the first
expression by xx -- 22
to get a common
denominator.
x − 3x + 2
x
2x - 1
x
2x - 1
+
=
+
x - 1 x 2 - 3x + 2 x - 1 (x - 1)(x - 2)
# xx -- 22 + (x -2x1)(x- 1- 2)
=
x
x-1
=
x 2 - 2x
2x - 1
+
(x - 1)(x - 2) (x - 1)(x - 2)
=
x 2 - 2x + 2x - 1
(x - 1)(x - 2)
=
x2 - 1
(x - 1)(x - 2)
=
(x - 1)(x + 1)
(x - 1)(x - 2)
Factor the denominators.
Rewrite each expression with
the LCD.
Add the numerators. Combine
like terms.
Factor the numerator and divide
out the common factors.
x+1
= x - 2, x ≠ 1
The sum of the expressions is xx +- 12 for x ≠ 1 and x ≠ 2.
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Problem 4
P
Subtracting Rational Expressions
What is the difference of the two rational expressions in simplest form?
State any restrictions on the variable. x2 + 2 − x + 2
S
How is this problem
similar to Problem 3?
The method is the same
except you subtract the
rational expressions
instead of adding them.
x − 2x
x+2
x+2
x+2
x+2
=
x 2 - 2x 2x - 4 x(x - 2) 2(x - 2)
Factor the denominators.
The
T LCD is 2x(x - 2).
The difference is
486
2x − 4
# 22 - 2(xx +- 22) # xx
=
x+2
x(x - 2)
=
2(x + 2)
x(x + 2)
2x(x - 2) 2x(x - 2)
=
2x + 4
x 2 + 2x
2x(x - 2) 2x(x - 2)
Simplify the numerators.
=
2x + 4 - (x 2 + 2x)
2x(x - 2)
Subtract the numerators.
=
- x2 + 4
2x(x - 2)
Combine like terms.
=
- (x2 - 4)
2x(x - 2)
Factor -1 from the numerator.
=
- (x - 2)(x + 2)
2x(x - 2)
Factor x 2 - 4 and divide out the
common factors.
=
- (x + 2)
2x
- (x + 2)
2x
for x ≠ 2 and x ≠ 0.
Lesson 11-5 Adding and Subtracting Rational Expressions
Rewrite each expression
with the LCD.
Problem 5
P
Simplifying a Complex Fraction
What is a simpler form of the complex fraction? State any restrictions on the variables.
x
1
x+y
1
y+1
What is the LCD of x1 ,
x
1
y , and y ?
The LCD of the rational
expressions is xy.
M
Method
1 Multiply both the numerator and the denominator by the LCD of all the
rational expressions and simplify the result.
( 1x + xy ) # xy
( 1y + 1 ) # xy
x #
1 #
x xy + y xy
= 1
#
#
y xy + 1 xy
1 x
x+y
=
1
y+1
y + x2
= x + xy
Multiply the numerator and
the denominator by xy.
Use the Distributive Property.
Simplify.
Method 2 Combine the expressions in the numerator and those in the denominator.
Then multiply the new numerator by the reciprocal of the new denominator.
#
#
1 x
1 y x x
x+y
x y+y x
= 1
y
1
y+1
y+1 y
How do you divide
a fraction by a
fraction?
Multiply the numerator
by the reciprocal of the
denominator.
#
Write equivalent expressions with
common denominators.
y
x2
xy + xy
=
1 y
y+y
Multiply.
y + x2
xy
= 1+y
y
Add.
=
y + x2 1 + y
xy , y
=
y + x2
xy
y + x2
= x + xy
# 1 +y y
Divide the numerator fraction by the
denominator fraction.
Multiply by the reciprocal.
Divide out the common factor, y.
The restrictions on the variables are x ≠ 0, and y ≠ 0.
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Problem 6
P
TEKS Process Standard (1)(A)
Using Rational Expressions to Solve a Problem
Fuel Economy A woman drives an SUV that gets 10 mi/gal (mpg). Her husband
drives a hybrid that gets 60 mpg. Every week, they travel the same number of
miles. They want to improve their combined mpg. They have two options on how
they can improve it.
Option 1:
They can tune the SUV and increase its mileage by 1 mpg and keep the
hybrid as it is.
Option 2:
They can buy a new hybrid that gets 80 mpg and keep the SUV as it is.
Which option will give them a better combined mpg?
The combined gas
mileage is total miles
divided by total gallons.
SUV miles + Hybrid miles
combined mpg = SUV gallons + Hybrid gallons
c
Let x = number of miles each drives in a week.
L
Define a variable and
describe each option.
Option 1
Tuned SUV gets 11 mpg.
Hybrid gets 60 mpg.
The gallons used by each
vehicle are miles
mpg . Write
the variable expressions
for each option’s
combined mpg.
Find the LCD of the
fractions in each
expression. Multiply
the numerator and
denominator by the LCD.
Distribute and simplify.
Round the ratios and
compare them.
488
x+x
x
+ 80
x+x
x
+ 60
x
10
x
11
2x
° x + x ¢
11
60
Option 2
SUV gets 10 mpg.
New hybrid gets
80 mpg.
# ( 660
660 )
2x
° x + x ¢
10
80
# ( 80
80 )
(2x)(660)
(2x)(80)
( 11x )(660) + ( 60x )(660)
( 10x )(80) + ( 80x )(80)
=
1320x
60x + 11x
=
160x
8x + x
=
1320x
71x
=
160x
9x
? 18.6 mpg
? 17.8 mpg
Option 1 gives the better combined mpg.
Lesson 11-5 Adding and Subtracting Rational Expressions
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Find
the least common multiple of each pair of polynomials.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. 9(x + 2)(2x - 1) and 3(x + 2)
2. x2 - 1 and x2 + 2x + 1
Simplify each sum or difference. State any restrictions on the variables.
d-3
1
1
3. 2x
+ 2x
- 5y
d-1
5. -x2 - 1x
4. 2d + 1 + 2d + 1
y+3
6. 2y - 1 - 2y - 1
5x
2
+
x2 - 9 x + 4
- 3x
12. 2
+ 4
x - 9 2x - 6
9.
9
3
+
x2 - 4 x2
y
3
10. 2y + 4 - y + 2
7.
13.
x
8
3x + 9 x 2 + 3x
1
1
x 2 - 25 x 2
5x
11. 2
+
x -x-6
2x
14. 2
x -x-2
8.
4
x 2 + 4x + 4
4x
x 2 - 3x + 2
15. Your car gets 25 mi/gal around town and 30 mi/gal on the highway.
a. If 50% of the miles you drive are on the highway and 50% are around town,
what is your overall average miles per gallon?
b. If 60% of the miles you drive are on the highway and 40% are around town,
what is your overall average miles per gallon?
Add or subtract. Simplify where possible. State any restrictions on the variables.
3
17. x + 1 + x - 1
7
4
+
x2 - 9 x + 3
20. 3x +
16. 4x - 22
x
19.
3
x
x 2 + 5x
x2 - 2
5x
x2 - x - 6
5y
21. 2
y - 7y 2y
18.
4
x 2 + 4x + 4
9
4
+
- 14 y
22. Apply Mathematics (1)(A) For the image of the overhead projector to be in focus,
the distance di from the projector lens to the image, the projector lens focal length
f, and the distance do from the transparency to the projector lens must satisfy the
thin-lens equation 1f = d1 + d1 . What is the focal length of the projector lens if the
i
o
transparency placed 4 in. from the projector lens is in focus on the screen located
8 ft from the projector lens?
23. Apply Mathematics (1)(A) To read small print, you
use a magnifying lens with the focal length 3 in. How
far from the magnifying lens should you place the
page? Use the thin-lens equation from Exercise 22.
24. Explain Mathematical Ideas (1)(G) Does the
Closure Property of rational numbers extend to
rational expressions? Explain and describe any
restrictions on rational expressions.
?
1 ft.
25. Explain Mathematical Ideas (1)(G) Explain how
factoring is used when adding or subtracting rational
expressions. Include an example in your explanation.
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Simplify each complex fraction. State any restrictions on the variables.
2 3
x+y
26. - 5 7
x +y
1
1
xy - y 2
28. 1
- 1
x 2y xy 2
1 + 2x
27.
3
2 + 2x
29.
2
x+4+2
3
1+x+4
30. Apply Mathematics (1)(A) The harmonic mean of two numbers a and b equals
2
1
1 . As you vary the length of a violin or guitar string, its pitch changes. If a full+
a b
length string is 1 unit long, then many lengths that are simple fractions produce
pitches that harmonize, or sound pleasing together. The harmonic mean relates
two lengths that produce harmonious sounds. Find the harmonic mean for each
pair of string lengths.
3
a. 1 and 12
3
3
b. 4 and 12
c. 4 and 5
d. 12 and 14
31. Show that the sum of the reciprocals of three different positive integers is greater
than 6 times the reciprocal of their product.
TEXAS Test Practice
T
32. Which expression equals 25x
x -9
-
4x
?
x2 + 5x + 6
A.
7x
(x - 3)(x + 3)(x + 2)
C.
x 2 + 22x
(x - 3)(x + 3)(x + 2)
B.
x 2 - 2x
(x - 3)(x + 3)(x + 2)
D.
9x 2 - 2x
(x - 3)(x + 3)(x + 2)
33. Which of the relationships is represented by the graph at the right?
F. y = log 4(x - 1) + 5
y
G. y = log 4(x - 1) - 2
2
O
H. y = log 4(x + 2) - 2
4
x
⫺2
J. y = log 4(x - 1) - 1
⫺4
2
x -5
34. What is a simpler form of 6
?
x -3
2 - 5x
2 + 5x
2x - 5
6 + 3x
B. 6 - 3x
C. 6x + 3
D. 2 - 5x
A. 6 - 3x
35. What word makes the statement “The domain and range of a(n) ________________
function is the set of all real numbers” sometimes true?
F. polynomial
G. logarithmic
H. exponential
J. quadratic
36. What is the least common denominator for the rational expressions 2 1
x - 5x - 6
1
and 2
? Show your work.
x - 12x + 36
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Lesson 11-5 Adding and Subtracting Rational Expressions