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When dealing with algebraic fractions, finding a common denominator is crucial for operations like addition and subtraction. A common denominator is simply a common multiple of the different denominators involved in the fractions you want to work with.
For algebraic fractions, we focus on multiplying the expressions in the denominators together to form this common base.
In our example, the denominators are \(x+1\) and \(x-1\). To find the common denominator, you multiply them to get \((x+1)(x-1)\). This shared denominator allows us to transform both fractions, making them comparable and easier to compute.

Once you've worked with the common denominator, the next key task is to simplify the resultant expression. Simplifying involves combining like terms and reducing the expression to its simplest form.
In the original task, once we have our fractions using the common denominator, the expression becomes quite simple.
You should always aim to cancel out any common factors in the numerator and the denominator, provided that these are not variables that could be zero or undefined.
To simplify the expression \(\frac{2x}{(x+1)(x-1)}\), you should first check for any common factors in the numerator and denominator. If there are no common factors, as in this case, then this is the simplest form of the expression. Remember to always simplify fractions unless told otherwise because it makes handling them much easier as you solve.

Adding and subtracting fractions requires that both fractions have the same denominator. Once you've found and applied a common denominator, the rest involves working with the numerators.
In our exercise, after aligning the fractions under one common denominator \((x+1)(x-1)\), the operation you need to perform involves simply adding the two numerators.

• Take the expression \((x-1) + (x+1)\). Combine like terms: \(x + x = 2x\) and the constants \(-1 + 1 = 0\).

Altogether, this results in \(2x\). With the common denominator in place, the subtraction or addition boils down to combining these modified numerators. This step consolidates the fractions into a single expression under one denominator.
Operate carefully to ensure there are no arithmetic errors. These small steps add up to simplify the more extensive process of working with algebraic fractions.