Problem 40 - Algebra 1

Question

SIMPLIFYING RATIONAL EXPRESSIONS Simplify the expression. $$\frac{3}{x}+\frac{x+9}{x}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{3}{x} + \frac{x + 9}{x}\) is \(\frac{x + 12}{x}\).

Step 1: Identify common denominator

Identify the common denominator of the two fractions. In this case, the common denominator for both fractions is \(x\).

Step 2: Simplify the expression

Since the denominators are the same, the two fractions can be combined by adding the numerators: \(\frac{3}{x} + \frac{x+9}{x} = \frac{3+x+9}{x}\).

Step 3: Simplify the numerator

The numerator simplifies to \(3+x+9\), which can be further simplified to \(x+12\). Therefore, \(\frac{3}{x} + \frac{x + 9}{x}\) simplifies to \(\frac{x + 12}{x}\).

Key Concepts

Common Denominator

When working with fractions, having a common denominator allows us to combine them with ease. Unlike adding or subtracting whole numbers, fractions need to be over the same base or denominator. The denominator is simply the number at the bottom of the fraction, which tells us how many equal parts the whole is divided into.

• If the denominators of two fractions are the same, it is straightforward to perform the addition or subtraction.
• If they aren’t, we need to find a common denominator before combining the fractions.

For our exercise, both fractions already have a denominator of \(x\). This simplifies the process because we can proceed straight to combining the numerators. There’s no need to adjust the fractions further, as the denominators are identical.

Combining Fractions

Once fractions have a common denominator, their numerators can be directly added or subtracted. Think of it like adding numbers — once the fractions are on an equal footing thanks to their common denominator:

• Simply add the numerators together and keep the denominator the same.
• Ensure the result remains as simple as possible by checking if further simplification is needed.

In the given problem, we add \(3\) and \(x+9\), which means we end up with \(3 + x + 9\) in the numerator. This step ensures our solution is both accurate and expressed in the simplest form possible.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations combined together. When simplifying algebraic expressions, it's essential to combine like terms and reduce them when possible. Here’s what we did:

• Identify and group like terms. In this case, \(3\) and \(9\) are constants we can combine.
• Manipulate expressions to make them simpler to understand and solve.

With our expression \(3 + x + 9\), we can combine \(3\) and \(9\) to get \(12\), resulting in \(x + 12\). By simplifying the numerator correctly, we end up with a cleaner expression, \(\frac{x + 12}{x}\), which is easier to interpret and use in further calculations. This method not only simplifies current problems but aids in understanding more complex algebraic work.