Problem 9 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \(\frac{x - 4}{5}\).

Step 1: Identify common factors

Look at the numerator to see if there is a common factor that can be factored out. The numerator is \(2x - 8\), and both terms in the numerator have a common factor of 2.

Step 2: Factor the common term

Factor 2 out of the terms in the numerator: \[2(x - 4)\]

Step 3: Rewrite the fraction

Rewrite the original fraction \(\frac{2x - 8}{10}\) using the factored form: \[\frac{2(x - 4)}{10}\]

Step 4: Simplify the fraction

Now simplify the fraction by canceling the common factor from the numerator and the denominator. The number 2 is common in both: \[\frac{2(x - 4)}{10} = \frac{2(x - 4)}{2 \times 5} = \frac{x - 4}{5}\]

Key Concepts

Factoring

Factoring is a technique used in algebra to simplify expressions. It involves breaking down a number or an expression into its composite parts, also known as factors. In the problem given, the numerator is \(2x - 8\). The first step is to identify common factors in this expression. Both 2x and 8 are divisible by the number 2. By factoring out the common term, we convert 2x - 8 into a simplified form: \[2(x - 4)\]. This process is crucial as it allows us to simplify more complex expressions with ease.

Numerator and Denominator

In a fraction, the numerator is the expression above the line, and the denominator is the expression below the line. For the fraction \( \frac{2x-8}{10} \), \(2x - 8 \) is the numerator, and 10 is the denominator. Understanding the roles of the numerator and denominator helps in simplifying fractions. In this example, we identified that both the numerator and the denominator can be factored. Once we factor the numerator, we rewrite the fraction as \[ \frac{2(x - 4)}{10} \].

Canceling Common Factors

Canceling common factors is a method used to simplify fractions. After factoring, we often find common factors in the numerator and the denominator that can be canceled out. In \[ \frac{2(x - 4)}{10} \], the common factor is the number 2. Division of both the numerator and the denominator by 2 simplifies the fraction: \[ \frac{2(x - 4)}{10} = \frac{2(x - 4)}{2 \times 5} = \frac{x - 4}{5} \]. This results in a simpler expression, \( \frac{x - 4}{5} \). This technique makes complex fractions much easier to manage.