Problem 42 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The simplified expression of \(\frac{2}{2 x}+\frac{12}{x}\) is \(\frac{13}{x}\).

Step 1: Simplify the Fractions

Before adding the fractions, it’s necessary to simplify them individually. The first fraction simplifies to \(\frac{1}{x}\), since dividing 2 by 2 leaves us with 1. The second fraction remains \(\frac{12}{x}\) as it’s already simplified.

Step 2: Find a Common Denominator

Both fractions \(\frac{1}{x}\) and \(\frac{12}{x}\) already have a common denominator, which is \(x\). Therefore, there is no need for further actions to find a common denominator.

Step 3: Add the Fractions

The fractions can now be added using the rule for adding rational expressions with a common denominator, \(a/b + c/b = (a+c)/b\). Therefore, \(\frac{1}{x} + \frac{12}{x}\) simplifies to \(\frac{1+12}{x}\), which is \(\frac{13}{x}\).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them is often the first step you should take. Simplifying a fraction means reducing it to its simplest form. For instance, if both the numerator (top number) and the denominator (bottom number) of a fraction can be divided by the same non-zero number, you should perform that division to simplify.

For example, in the expression \(\frac{2}{2x}\), both the 2s in the numerator and denominator can be divided by 2, which gives us \(\frac{1}{x}\). Simplifying ensures that the expression is in its simplest terms, making further operations like addition or subtraction easier down the road.

Common Denominator

Before adding or subtracting fractions, you need a common denominator. The denominator is the bottom part of a fraction, and it tells us into how many pieces the whole is divided. Having a common denominator means both fractions are equally divided, allowing you to directly add or subtract the numerators.

In our given problem, the fractions \(\frac{1}{x}\) and \(\frac{12}{x}\) already have a common denominator of \(x\). This makes our task easier since we don’t need to do anything else to align the fractions for addition.

• A common denominator allows for the straightforward manipulation of fractions.
• This step is crucial in ensuring that fractions are in a form that can easily be added or subtracted.

Adding Fractions

Adding fractions might seem daunting at first, but it's quite simple with a common denominator. When you add fractions that have the same denominator, you only need to add the numerators. The denominator stays the same.

Consider the expression we simplified earlier: \(\frac{1}{x} + \frac{12}{x}\). Both fractions share the same denominator of \(x\), so we sum the numerators: 1 and 12. This gives us 13, and the resulting fraction is \(\frac{13}{x}\).

• Remember to only add the numerators when the denominators are the same.
• Keep the denominator unchanged to maintain the common division of the whole.

Following these steps carefully ensures that you arrive at the correct simplified result.