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To perform the subtraction of rational expressions, the first step is finding the least common denominator (LCD). The LCD is the smallest expression that is a multiple of each individual denominator.
For instance, take the example from our exercise: \(\frac{1}{x-1} - \frac{1}{x}\).

The denominators here are \(x-1\) and \(x\).
To find the LCD, look for the product of these denominators since they don't share common factors other than 1.
Multiplying them together gives: \(x(x-1)\).
This becomes our least common denominator, which means both fractions need to be rewritten using this denominator.

Identifying the LCD is crucial because it standardizes the denominators, allowing us to easily add or subtract the numerators.

A rational expression is a fraction in which both the numerator and the denominator are polynomials. In our exercise, both \(\frac{1}{x-1}\) and \(\frac{1}{x}\) are rational expressions.
Rational expressions behave similarly to regular fractions, so many of the same principles and rules apply.

Here are a few key points to remember about rational expressions:

• They can become undefined if the denominator equals zero.
• You can only add or subtract rational expressions when they have the same denominator.
• Multiplying and dividing rational expressions is straightforward but requires factoring and simplifying.

Recognizing and working with rational expressions is an important skill in algebra because they appear frequently in various forms, especially in more complex equations and calculus.

Subtracting fractions, and specifically rational expressions, follows a clear process. First, ensure both fractions share a common denominator.
In our exercise: \(\frac{1}{x-1} - \frac{1}{x}\), we determined the LCD to be \(x(x-1)\).

With this common denominator, the fractions can be rewritten:

• \(\frac{1}{x-1} = \frac{x}{x(x-1)}\)
• \(\frac{1}{x} = \frac{x-1}{x(x-1)}\)

Once the denominators match, subtract the numerators:
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