Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{x}{2}+\frac{5}{2} $$

Short answer

\( \frac{x + 5}{2} \)

Step-by-step solution

Identify the Given Terms

Identify the terms in the given expression. The expression given is \( \frac{x}{2} + \frac{5}{2} \).

Combine the Fractions

Since both fractions have the same denominator, combine them by adding the numerators together. This results in \( \frac{x + 5}{2} \).

Simplify the Expression

Since there are no common factors in the numerator beyond what's already given, the simplified form of the expression is \( \frac{x + 5}{2} \).

Key concepts

adding fractions

Adding fractions can sometimes seem tricky, especially if you're not confident on the steps. Let’s break it down. When the denominators (the bottom numbers) are the same, all you need to do is add the numerators (the top numbers). The denominator stays the same.

For example, in the expression \(\frac{x}{2}+\frac{5}{2}\), both fractions have a denominator of 2.

To add these fractions, you simply add the numerators (\x\ + 5) and keep the denominator (2), leading us to \(\frac{x + 5}{2}\).

Easy, right? You can apply the same idea to any fractions with the same denominator. Here are the steps:

• Ensure the denominators are the same
• Add the numerators
• Keep the denominator unchanged

Remember, if the denominators aren't the same, you'll need to find a common denominator before you can add the numerators.

factored form

Factored form means expressing an expression as a product of its factors rather than as a sum. It is a way of breaking down an equation into simpler pieces (factors) that, when multiplied together, give you the original expression.

In the given exercise, the final expression is \(\frac{x + 5}{2}\). Here, \x + 5\ is a binomial that is already simplified and factored as much as it can be.

To leave it in factored form:

• Look at each term in the numerator and see if they can be factored further.
• Identify any common factors in the numerator.

In our example of \(\frac{x + 5}{2}\), \x + 5\ cannot be factored further because \x\ and 5 have no common factors.

So, the expression is already in its simplest factored form.

simplifying expressions

Simplifying expressions means reducing an algebraic expression to its simplest form. This involves combining like terms and factoring wherever possible.

To simplify \(\frac{x + 5}{2}\):

• First, combine any like terms in the numerator. In our exercise, the numerator is already simplified as there are no like terms to combine.
• Next, check for any common factors in the numerator that can be factored out. Here, since \x + 5\ cannot be factored further and there are no common factors with 2, it remains as is.

This process ensures the expression is as simplified as possible. Simplifying expressions makes it easier to work with them in more complex mathematical operations.