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To add or subtract fractions, it's crucial to have a common denominator. The denominator is the bottom number of a fraction, and it indicates into how many equal parts the whole is divided.

When fractions have different denominators, you can't directly add or subtract them. Instead, you need to find a common denominator - a shared multiple of both denominators.

For example, in our exercise, the denominators are \(x\) and \(4x\). We need a number that both \(x\) and \(4x\) can divide without leaving a remainder. That number is \(4x\).

Finding a common denominator helps convert fractions to an equivalent form where the denominators are the same, making addition or subtraction straightforward.

Once you have a common denominator, you can proceed with adding the fractions. Let's consider our example where we converted \( \frac{5}{x} \) to \( \frac{20}{4x} \). Notice that \( \frac{9}{4x} \) already has the common denominator.

To add fractions with the same denominator, simply add their numerators while keeping the denominator unchanged. This results in:

\( \frac{20}{4x} + \frac{9}{4x} = \frac{20 + 9}{4x} = \frac{29}{4x} \).

With a common denominator in place, fraction addition becomes as simple as adding the top numbers (numerators) together.

Simplifying a fraction means reducing it to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

However, in our example, the final fraction we obtained was \( \frac{29}{4x} \). Here, \(29\) and \(4x\) don't have any common factors other than 1, so the fraction is already in its simplest form.

This step may not always be necessary, but it's important to recognize when a fraction can be simplified. This makes the fraction easier to understand and work with in future calculations.