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When we add fractions, we need a common denominator. For fractions with different denominators, the least common denominator (LCD) is essential. It is the smallest expression that can contain each denominator without leaving any remainder.
To find the LCD for

• \(\frac{6}{x+4}\)
• \(\frac{3}{x-1}\)

look at the denominators \((x+4)\) and \((x-1)\).
The best way to combine them is to multiply them together, creating a common base for both fractions. This means the LCD is \((x+4)(x-1)\), enabling us to rewrite both fractions using this denominator. Once you rewrite both fractions with the same denominator, they can easily be added.

Simplifying fractions is all about finding the simplest form of a fraction, where the numerator and denominator have no common factors besides 1. After rewriting fractions using the least common denominator, you may notice that the numerators become expressions such as

• \(6(x-1)\)
• \(3(x+4)\)

These should be expanded to

• \(6x - 6\)
• \(3x + 12\)

After expanding, the fraction is grouped over the LCD \((x+4)(x-1)\).
The next step towards simplifying further involves checking if there’s a way to cancel out any common factors in both the numerator and denominator. In this example, the denominator has no common factors with the numerator that simplify down any further, so ensure you write down the simplified version to visualize better without any of those larger numbers distracting you!

Combining like terms is a crucial algebraic technique to make expressions simpler and more manageable. After rewriting our complex fractions and expanding the numerators, it’s time to consolidate those expressions.
When you have

• Expressions like \(6x - 6\) and \(3x + 12\)

combine them by adding the coefficients of similar terms.
Identify the terms with similar variables, in this case, terms involving \(x\), namely \(6x\) and \(3x\). Add these together to get \(9x\).
Similarly, tackle the constants, namely \(-6\) and \(+12\), which sum up to \(+6\).
So, the expression becomes \(9x + 6\). All of this lies above the common denominator \((x+4)(x-1)\), providing you the final fraction \(\frac{9x+6}{(x+4)(x-1)}\).
Combining like terms simplifies the problem, resulting in a cleaner, understandable expression.