The length of a garden plot is . The width of the plot is . How much longer is the garden plot than it is wide?

### Guidance

Recall when two fractions do not have the same denominator. You have to multiply one or both fractions by a number to create equivalent fractions in order to combine them.

Here, 2 goes into 4 twice. So, we will multiply the first fraction by to get a denominator of 4. Then, the two fractions can be added.

Once the denominators are the same, the fractions can be combined. We will apply this idea to rational expressions in order to add or subtract ones without like denominators.

#### Example A

Subtract .

**
Solution:
**
Factoring the denominator of the first fraction, we have
. The second fraction needs to be multiplied by
in order to make the denominators the same.

Now that the denominators are the same, subtract the second rational expression just like in the previous concept.

The numerator is not factorable, so we are done.

#### Example B

Add

**
Solution:
**
Factoring the second denominator, we have
. So, we need to multiply the first fraction by
.

#### Example C

Subtract .

**
Solution:
**
Factoring the first denominator, we have
. This is the Lowest Common Denominator, or LCD. The second fraction needs the 2 and the
.

**
Intro Problem Revisit
**
We need to subtract the width from the length.

Factoring the first denominator, we have . So, we need to multiply the second fraction by .

Therefore, the garden plot is longer than it is wide.

### Guided Practice

Perform the indicated operation.

1.

2.

3.

#### Answers

1. The LCD is or . Multiply the first fraction by .

2. Here, the LCD or . Multiply the second fraction by .

3. The LCD is . The second fraction’s denominator factors to be , so it needs to be multiplied by .

### Practice

Find the LCD.

Perform the indicated operations.