Problem 11 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{2 x-8}{10 x} $$

Step-by-Step Solution

Answer

\( \frac{x - 4}{5x} \)

Step 1: Factor the Numerator

First, factor out the greatest common factor (GCF) from the numerator. The numerator is \(2x - 8\). The GCF of \(2x\) and \(-8\) is 2. So, factor 2 out of the numerator: \(2(x - 4)\).

Step 2: Simplify the Expression

Rewrite the fraction with the factored numerator: \[ \frac{2(x - 4)}{10x} \]. Next, divide both the numerator and the denominator by their GCF to simplify the fraction. The GCF of 2 and 10 is 2. Thus, \[ \frac{2 \times (x - 4)}{2 \times 5x} = \frac{x - 4}{5x} \].

Key Concepts

greatest common factor

The greatest common factor (GCF) is a crucial concept in simplifying algebraic fractions. It refers to the highest number that can evenly divide both terms of an expression.
For example, consider the expression in the numerator: \(2x - 8\).
The terms here are \(2x\) and \(-8\). To find the GCF, let's focus on the factors each term shares.

• For \(2x\), the factors are 2 and x.
• For \(-8\), the factors are 2 and 4.

The common factor is 2, making it the GCF. By factoring out the GCF, you simplify the expression and make further calculations easier.

factoring

Factoring is the process of breaking down an expression into simpler parts that can be multiplied together to get the original expression.
When you factor, you're looking to rewrite the expression in a more manageable form. In the given problem, the goal is to factor the numerator \(2x - 8\).
Using the GCF we identified, you can factor out 2 from each term:

• \[2x = 2 \times x\]
• \[-8 = 2 \times (-4)\]

Thus, the expression factors to:
\[2(x - 4)\]
This step simplifies the fraction by making it clearer what can be canceled out with the denominator.

simplifying expressions

Simplifying expressions means reducing them to their most manageable form. Once you've factored the numerator in \(2(x-4)\), the next step is to simplify the entire fraction: \frac{2(x - 4)}{10x}\.
Look for common factors in the numerator and the denominator. In this case, the GCF of 2 (from the numerator) and 10 (from the denominator) is 2.
Now, divide both the numerator and the denominator by 2 to reduce the fraction:

• Numerator: \(2(x-4) \rightarrow (x-4)\)
• Denominator: \(10x \rightarrow 5x\)

This gives you: \frac{x-4}{5x}\
The fraction is now fully simplified. Remember, simplifying expressions often involves several steps like finding the GCF, factoring, and canceling common factors.