[MODULE 6 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS]
New York City College of Technology
MAT 1175 PAL Workshops
Name: ___________________________________
Points: ______
Adding and Subtracting Rational Expressions
1. Adding and Subtracting Rational Expressions with the Same Denominators
Recall in fractions if the denominators are the same:
1 3 1 3 4 1
 
 
8 8
8
8 2
To add or subtract rational expressions with the same denominators:
1. Add or subtract the numerators.
2. Write the result over the common denominator.
3. Simplify the rational expression, if necessary.
a) Add:
c) Subtract:
5y
4y

y9 y9
b) Add:
7
6
 2
2
r
r
3 x  11 x  3

x2
x2
d) Subtract:
x2
25

x 5 x 5
2. Finding the Least Common Denominator (LCD) of Rational Expressions
To find the Least Common Denominator (LCD) of rational expressions:
1. Factor each denominator completely.
2. The least common denominator (LCD) is the product of all unique factors each raised to the greatest
power that appears in any factored denominator.
a) Find the LCD for the rational expressions:
2
5z
,
5 2
3x y 4 xy 4
b) Find the LCD for the rational expressions:
8
x
,
x 5 x 5
1
Prepared by Profs. Ghezzi, Han, and Liou-Mark
Supported by BMI, NSF STEP #0622493, and Perkins
[MODULE 6 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS]
New York City College of Technology
MAT 1175 PAL Workshops
c) Find the LCD for the rational expressions:
y 1
3y
7
, 2
, 2
y  25 2 y  9 y  5 y  10 y  25
2
3. Adding and Subtracting Rational Expressions with Different Denominators
Recall in fractions if the denominators are different:
1 3 1 4 3 3 4
9 13
1
      


1
3 4 3  4  4  3  12 12 12
12
To add and subtract rational expressions with different denominators
1. Find the LCD of the rational expressions.
2. Write each rational expression as an equivalent rational expression whose denominator
is the LCD found in step 1.
3. Add or subtract the numerators and write the result over the common denominator.
4. Simplify the resulting rational expression.
a)
c)
4
3

3m 2m 2
LCD = _______________
7
a
 2
a  a  2 a  4a  3
2
2
b)
3
8

2 y  10 3 y  15
LCD = ________________________
Prepared by Profs. Ghezzi, Han, and Liou-Mark
Supported by BMI, NSF STEP #0622493, and Perkins
LCD = _____________
[MODULE 6 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS]
New York City College of Technology
MAT 1175 PAL Workshops
d)
x  3 x 1

x 5 x 5
LCD = ___________________
e)
x 1
3
 2
x  6 x  8 x  16
LCD = ___________________
2
4. What is the difference between the following two problems:
Problem 1: Add or subtract as indicated, simplify the answer:
Problem 2: Solve the rational equation:
3
Prepared by Profs. Ghezzi, Han, and Liou-Mark
Supported by BMI, NSF STEP #0622493, and Perkins
1
2
1

 2
a3 a3 a 9
1
2
1

 2
a3 a3 a 9
[MODULE 6 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS]
New York City College of Technology
MAT 1175 PAL Workshops
5. Solve each equation.
a.
2 5 1
 
x 6 4x
c.
3
e.
2x
1
1
 
2x  1 x 2x  1
4
5 2

0
x x2
Prepared by Profs. Ghezzi, Han, and Liou-Mark
Supported by BMI, NSF STEP #0622493, and Perkins
b.
2y  7 y 1 y  3


9
6
4
d.
3
1
4

 2
2x  3 2x  3 4x  9
f.
12
1
 1
x4
3 x  12 x
2