Problem 67 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{2}{2 x}+\frac{12}{x} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{26}{x}\).

Step 1: Identify the Common Denominator

To simplify the expression and add these fractions together, we need to find a common denominator. The common denominator here is \(x\).

Step 2: Express Both Fractions With the Common Denominator

The fractions are already expressed in terms of the common denominator \(x\). So, no changes are needed here.

Step 3: Add the Fractions

Once the fractions have the same denominator, we can add the numerators. So, \(\frac{2}{2x} + \frac{12}{x} = \frac{2+24}{x}\).

Step 4: Simplify the Expression

The last step is to simplify the expression we have obtained. Thus, the simplified form of the given expression is \(\frac{26}{x}\).

Key Concepts

Common Denominator

Understanding the concept of a common denominator is crucial when working with algebraic fractions. It's the key to combining fractions in a way that makes them easier to manage. Think of the common denominator as a shared ground for fractions, much like a common language allows people to communicate effectively.

In algebra, the common denominator is the product of the different denominators of the fractions being added or subtracted, after factoring out any common factors. In the given exercise, we identify that the fractions have different denominators, one being 2x and the other x. Since every number is divisible by 1, and x can divide 2x, we recognize that our common denominator is simply x.

Adding Fractions

When adding fractions, if they don't already have the same denominator, we need to find the common denominator before we proceed. It's like syncing two clocks to make sure they tell the same time before comparing. Once we have the common denominator, we add only the numerators (the top numbers of the fractions) while keeping the denominator the same.

This is what we did in our exercise: both fractions were expressed with the common denominator x, so they were ready to be added directly. We simply added the numerators 2 and 24 to get 26, while x stayed as our common denominator.

Algebraic Fractions

Dealing with algebraic fractions can be a bit like working with a puzzle where variables and numbers are intermingled. An algebraic fraction includes variables (like x), making it more complex than a simple numerical fraction. In the original problem, we dealt with algebraic fractions like \( \frac{2}{2x} \) and \( \frac{12}{x} \).

The complexity of algebraic fractions comes from the additional step of accounting for the variable when performing operations like addition, subtraction, multiplication, or division. It's important to remember that the same rules for numerical fractions apply; we use common denominators to add or subtract and multiply or divide the numerators and denominators just as with integers.

Simplify Expression

To simplify an algebraic expression means to make it as straightforward as possible. It involves combining like terms, reducing fractions by eliminating common factors, and performing any possible arithmetic operations. With fractions, once you've successfully added them by using a common denominator, the next step is to simplify.

In our exercise, after adding the fractions, we were left with \( \frac{26}{x} \). This expression cannot be simplified further, as 26 and x have no common factors other than 1. Screwing in the final bolt in our construction, we confirm our structure is sound—the expression is as simple as it gets.