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

• 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

• 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

3

3x

Write each rational as an equivalent expression with the LCD

2

2

x

x

3 2x x

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

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

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

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

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.

• 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

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

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.

• 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

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

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.

• 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

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

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

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