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When adding fractions, determining whether they have a common denominator is the first and most crucial step. The denominator is the bottom part of the fraction that tells us into how many equal parts the whole is divided. If the denominators are the same, the process is significantly simplified. It allows you to combine the numerators directly without altering the denominator.
You only have to focus on the top numbers, or numerators, since the bottom numbers, or denominators, are already aligned. So when you see a problem like \( \frac{x}{x-2} + \frac{3}{x-2} \), notice that both fractions have \( x-2 \) as a denominator. This tells us we can proceed directly to the next step - adding the numerators.

Numerator addition is straightforward when fractions share common denominators. Simply add the numerators together. Imagine you have a parking lot with several spots (your denominator), each occupied by several cars (your numerators). Adding the numerators means counting all the cars together.
In our example, the numerators are \( x \) and \( 3 \). When the denominators are equal, we add the numerators: \( x + 3 \). Ensure that while you are performing this arithmetic operation, the denominator of the fractions remains unchanged. So, you effectively have a single fraction with the numerator \( x + 3 \) over the unchanged denominator \( x-2 \). This results in \( \frac{x+3}{x-2} \).

Once you've added the numerators, the resulting fraction is almost complete. However, before finalizing your answer, it's essential to check if the expression can be further simplified. A simplified expression means reducing the fraction to its simplest form, where both the numerator and the denominator have no common factors other than 1.
In this particular case \( \frac{x+3}{x-2} \), examine if the numerator \( x+3 \) and the denominator \( x-2 \) share any common factors. Since there are none here, \( \frac{x+3}{x-2} \) is already as simplified as it can get.
This step often involves factoring both the numerator and the denominator to see if any terms cancel out, but be cautious not to oversimplify, as it might alter the original meaning of the expression.