Problem 12 - Algebra 1: Concepts and Skills

Question

ADDING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{7}{2 x}+\frac{x+2}{2 x} $$

Step-by-Step Solution

Answer

The simplified expression of \(\frac{7}{2x} + \frac{x+2}{2x}\) is \(\frac{x+9}{2x}\).

Step 1: Identify the Denominators

In this case, both fractions already have the same denominator: \(2x\). So, no need to find a common denominator.

Step 2: Add the Numerators

Since both fractions have the same denominator, just add the numerators \((7 + (x+2))\) to get the result.

Step 3: Simplify the Result

After adding, we obtain the fraction \(\frac{7 + x + 2}{2x}\). This simplifies to \(\frac{x + 9}{2x}\).

Key Concepts

Simplify Expressions

When tackling algebraic problems, simplifying expressions is often a critical first step. Simplification makes expressions more manageable and solutions clearer. Take the expression \(\frac{7}{2x} + \frac{x+2}{2x}\). Here, simplification involves combining like terms and reducing fractions whenever possible. It's much like tidying up - combine scattered items that belong together (like terms) and throw out anything unnecessary (reducing fractions). For this particular problem, after adding the numerators, we get \(\frac{7 + x + 2}{2x}\), which simplifies neatly into \(\frac{x + 9}{2x}\), much like organizing a cluttered room into a neat and orderly space.

Common Denominator

In the realm of algebraic fractions, the 'common denominator' is a bit like a common language for fractions to communicate. It allows fractions to be added or subtracted by giving them an identical base to stand on. Finding a common denominator often involves identifying the least common multiple of the different denominators. In our exercise, both fractions already share a common denominator of \(2x\), which means they're already prepped for easy communication - no translation needed! When you encounter expressions with different denominators, remember that aligning them to a common denominator is like getting everyone on the same page before proceeding with calculations.

Algebraic Fractions

Algebraic fractions are simply fractions that include variables, like \(\frac{7}{2x}\) and \(\frac{x+2}{2x}\). They follow all the same rules as regular fractions but are slightly more complex due to the addition of letters to numbers. Think of algebraic fractions as ordinary fractions' more sophisticated siblings - they may require a bit more effort to understand at first, but once you get the hang of them, they follow the same patterns. Adding, subtracting, multiplying, and dividing them is just like dealing with numerical fractions, but remember to pay special attention to simplifying and factoring when dealing with the variable parts.

Numerators Addition

Numerators addition is exactly what it sounds like: adding the top parts, or numerators, of fractions. It's a simple concept that's executed easily when the denominators are the same, as they are in our exercise. Imagine the numerators are pieces of fruit in a basket – to find out how much fruit you have, add them together as long as they're in the same basket (denominator). In our example, adding \(7\) and \(x+2\) gives us \(x + 9\), all neatly contained within the \(2x\) basket. This process is the backbone of combining fractions and is an essential skill in algebra. Just remember, when the denominators differ, you'll need to find a common denominator before you can start adding the numerators. Think of it as getting all the fruit into identical baskets before you can count them up.