91Ó°ÊÓ

When dealing with fractions, especially when subtracting them, finding a common denominator is key. A common denominator is simply a shared multiple of the original denominators. In this exercise, you have two fractions: \( \frac{1}{x+1} \) and \( \frac{1}{x+2} \).

To subtract these fractions, both must have the same denominator. The easiest way to find a common denominator is to multiply each denominator together, in this case, \((x+1)(x+2)\).

This product represents the least common denominator (LCD), which is the smallest expression that both original denominators can divide into without leaving a remainder.

This step is crucial because it allows you to convert the fractions into equivalent fractions with the same denominator. Once they share a common denominator, you can easily perform operations like subtraction.

Subtracting fractions may seem complex, but it's quite straightforward once you have a common denominator. After rewriting each fraction to have the same denominator of \((x+1)(x+2)\), you simply subtract the numerators.

In our exercise, this means your fractions will become \( \frac{x+2}{(x+1)(x+2)} \) and \( \frac{x+1}{(x+1)(x+2)} \). The subtraction process involves only the numerators, not the denominators.

So you subtract the numerators: \((x+2) - (x+1)\), which equals 1. The denominator remains the same, leading to the fraction \( \frac{1}{(x+1)(x+2)} \).

Important to remember:

• Ensure both fractions have the same denominator before subtracting.
• Keep the common denominator unchanged while performing operations on numerators.

Once subtraction is performed, it's important to simplify the resulting fraction if possible. Simplifying a fraction involves making it as straightforward as possible by reducing it to its simplest form.

In our exercise, after subtracting the numerators and obtaining \( \frac{1}{(x+1)(x+2)} \), you check if the numerator and denominator have any common factors.

Here are some tips for simplifying:

• Inspect the numerator and denominator for any common factors that can be canceled out.
• Remember that a fraction is in its simplest form when you cannot divide the numerator and denominator by any number other than 1.
• If no common factors are found, it means the fraction is already simplified, as with our exercise's final result \( \frac{1}{(x+1)(x+2)} \).

Since the numerator is 1, and the denominator is a polynomial expression with no common factors, this fraction is already in its simplest form, concluding the simplification process.