Table of Contents

Add & Subtract Rationals – Unlike Denominators

• It is assumed that you understand the material in the three previous modules:

1) Addition and subtraction of rational expressions with common denominators

2) Finding the LCM of two expressions

3) Writing equivalent rational expressions

Table of Contents

• To add or subtract rational expressions with unlike denominators:

1) Determine the LCD (LCM of the denominators)

2) Write each rational expression as an equivalent rational expression with the LCD as its denominator

3) Add or subtract as you did with common denominators

Table of Contents

• Example 1

Determine the LCD

3 8

3 2x x

Add the two rational expressions

3 2x x

Note that the denominators are not common

Table of Contents

3

3x

Write each rational as an equivalent expression with the LCD

2

2

x

x

3 2x x

Table of Contents

3

3x

Write each rational as an equivalent expression with the LCD

2

2

x

x

3 2

3 2x

x

x

3 6

3 2x

x

x

8

2x 3

3

x

x

3 2x x

Table of Contents

3

3x

Write each rational as an equivalent expression with the LCD for the denominator

2

2

x

x

3 2

3 2x

x

x

3 6

3 2x

x

x

8

2x 3

3

x

x

8 3

3 2x

x

x

8 4

3 2

2

x

x

x

Table of Contents

Add the equivalent rational expressions that now have a common denominator

3 6

3 2

x

x x

8 24

3 2

x

x x

3 8

3 2x x

Table of Contents

3 6

3 2

x

x x

8 24

3 2

x

x x

3 6 8 24

3 2

x x

x x

3 6 8 24

3 2

x x

x x

Table of Contents

3 6 8 24

3 2

x x

x x

11 18

3 2

x

x x

Since we can’t factor the numerator and nothing can reduced, this is the answer.

Table of Contents

• Example 2

3 8

3 2x x

In the last problem, we went to great detail to find the equivalent rational expressions. Here is a shorter version of the same problem.

Add the two rational expressions

Determine the LCD 3 2x x

Table of Contents

Write the equivalent rational expressions

3 2 3 2x x x x

3 8

3 2x x

2

2

x

x

3 6x

3

3

x

x

8 24x

Table of Contents

3 6 8 24

3 2

x x

x x

11 18

3 2

x

x x

Since we can’t factor the numerator and nothing can be reduced, this is the answer.

Table of Contents

• Example 3

2 3 4

2 5

10 12x y x ySubtract the two

rational expressions

Note that the denominators are not common

Determine the LCD

2 3 2 310 2 5x y x y 4 2 412 2 3x y x y

2 4 32 3 5 x y 4 360x y

Table of Contents

2 3 4

2 5

10 12x y x y

4 3 4 360 60x y x y

Write the equivalent rational expressions2

2

6

6

x

x

212x

2

2

5

5

y

y

225y

Table of Contents

2 2

4 3 4 3

12 25

60 60

x y

x y x y

2 2

4 3

12 25

60

x y

x y

Since we can’t factor the numerator and nothing can be reduced, this is the answer.

Table of Contents

• Example 4

2 2

2

5 6 3 2

x

x x x x

Subtract the two rational expressions

Note that the denominators are not common

Determine the LCD

Table of Contents

2 2

2

5 6 3 2

x

x x x x

2 3x x 1 2x x

2 3 1 2 3 1x x x x x x

1

1

x

x

1x x

3

3

x

x

2 3x

Table of Contents

1 2 3

2 3 1

x x x

x x x

2 2 6

2 3 1

x x x

x x x

2 6

2 3 1

x x

x x x

Be careful of the sign here

Table of Contents

2 6

2 3 1

x x

x x x

3 2

2 3 1

x x

x x x

3

3 1

x

x x

Table of Contents