Elementary Algebra
Test #7 Review
Page 1 of 6
Section 7.1: The Fundamental Property of Rational Expressions
To simplify rational expressions:
1. Factor the numerator.
2. Factor the denominator.
3. Cancel any common factors.
Example:
 x  1 x  1
x2 1

2
x  2 x  1  x  1 x  1

 x  1
 x  1
Section 7.2: Multiplying and Dividing Rational Expressions
P R PR
To multiply rational expressions:  
Q S QS
1. The numerator of the answer is the product of the numerators.
2. The denominator of the answer is the product of the denominators.
3. Hint: it helps to write everything in factored form, because then it is easy to see what cancels in your
final answer.
Example:
 x  1 x  6   6  x  3
x2  7 x  6
6 x  18
 2

3x  6
x  2 x  15
3 x  2
 x  5 x  3

6  x  1 x  6  x  3
3  x  2  x  5  x  3

6  x  1 x  6 
3  x  2  x  5 
Elementary Algebra
Test #7 Review
Page 2 of 6
P R P S PS
   
Q S Q R QR
1. Multiply the first rational expression times the reciprocal of the second rational expression.
To divide rational expressions:
Example:
a b  a 
ab  a 2
a b
a b
 2


2
a  1 a  2a  1  a  1 a  1  a  1 a  1

a  a  b   a  1 a  1

 a  1 a  1  a  b 

a  a  b  a  1 a  1
 a  1 a  1 a  b 

a  a  1
a 1
Section 7.3: Least Common Denominators
To write rational expressions with least common denominators:
1. Factor each denominator into prime factors.
2. List each different denominator factor the greatest number of times it appears in any one of the
denominators.
3. Multiply numerator and denominator of each given rational expression by factors from the list that do
not occur in the denominator of that given rational expression.
Example:
1
1
z 1
z 1



2
z  4 z z  z  4  z  1 z  z  4  z  1
4
4
z
4z

 
z  3z  4  z  4  z  1 z z  z  4  z  1
2
Elementary Algebra
Test #7 Review
Page 3 of 6
Section 7.4: Adding and Subtracting Rational Expressions
Adding Rational Expressions with the Same Denominator:
P
R
P R PR
If
and
are rational expressions, then  
. That is to say, to get the answer simply put the sum
Q Q
Q
Q
Q
of the numerators over the common denominator.
Adding (or Subtracting) Rational Expressions with Different Denominators:

Find the least common denominator (LCD).

Rewrite each rational expression as an equivalent expression with the LCD as the denominator.

Put the sum (or difference) of the numerators over the LCD.

Simplify the answer using the fundamental property of rational expressions.
Example:
2k
3
2k
3
 2


2
k  5k  4 k  1  k  4  k  1  k  1 k  1

2k
k 1
3
k 4



 k  4  k  1 k  1  k  1 k  1 k  4

2k  k  1  3  k  4 
 k  4  k  1 k  1

2k 2  2k  3k  12
 k  4  k  1 k  1

2k 2  5k  12
 k  4  k  1 k  1

2k 2  2k  3k  12
 k  4  k  1 k  1

 2k  3 k  4 
 k  4  k  1 k  1
Elementary Algebra
Test #7 Review
Page 4 of 6
Section 7.5: Complex Fractions
Method #1 To simplify a complex fraction:
1. Simplify the numerators and denominators into single fractions.
2. Change the complex fraction into a division problem.
3. Invert the second fraction and multiply.
Examples:
1
2
1
1
2
2 3 
2
4 3 1
4
4 1

 2 2
12 1

4 4
5
 2
13
4
5 13
 
2 4
5 4
 
2 13
10

13
1
x 1

x  x x
1 x2 1
x

x
x x
x 1
 2x
x 1
x
x  1 x2  1


x
x
x 1 x


x x2 1
 x  1  x

x   x  1 x  1
1

1
x 1
Elementary Algebra
Test #7 Review
Page 5 of 6
Section 7.6: Solving Equations with Rational Expressions
To solve an equation with rational expressions:
1. Multiply each side of the equation by the LCD.
2. Solve the resulting equation.
3. Check your answers by plugging them back in to the original equation. Remember that zero
denominator answers are not allowed.
Example:
6
1
4

 2
5a  10 a  5 a  3a  10
6
1
4


5  a  2  a  5  a  5  a  2 
6
1
4
 5  a  5  a  2  
 5  a  5  a  2  
 5  a  5  a  2 
5  a  2
a 5
 a  5  a  2 
6  a  5  5  a  2   4  5
6a  30  5a  10  20
a  40  20
a  60
Section 7.7: Applications of Rational Expressions
To solve word problems:
1. Draw a picture.
2. Label the picture with the unknowns and the given information.
3. Write an equation that relates all of the information.
4. Solve the equation.
5. Check your answer.
Special equations:
1. Distance, Rate, and Time Relationship: d = rt.
2. Rate of Work: If a job can be done in t units of time, then the rate of work is:
1
job per unit of time.
t
Elementary Algebra
Test #7 Review
Page 6 of 6
Section 7.8: Variation
Direct Variation:
1. y varies directly as x means the equation: y = kx.
2. Or, we also say y is proportional to x.
3. k is called the constant of variation (or proportionality constant).
Direct Variation as a Power:
1. y varies directly as the nth power of x means the equation: y = kxn.
Inverse Variation:
1. y varies inversely as x means the equation: y 
k
x
2. y varies inversely as the nth power of x means the equation: y 
k
xn