Problem 77 - Intermediate Algebra

Question

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{2 x-14}{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(x - 7\).

Step 1: Factor the Numerator

The numerator is given as \(2x - 14\). This expression can be factored by taking out the greatest common factor (GCF), which is 2. Factoring out 2, the expression becomes \(2(x - 7)\). So the factored numerator is \(2(x - 7)\).

Step 2: Factor the Denominator

The denominator is \(2\). Since it is already a prime number and doesn't have any factors other than itself, it remains as \(2\).

Step 3: Simplify the Expression

Now, we have the fraction \(\frac{2(x - 7)}{2}\). We can simplify this by canceling out the common factor of 2 from both the numerator and the denominator. After canceling, we are left with \(x-7\).

Key Concepts

Factoring Expressions

Factoring expressions is a key skill in algebra that involves breaking down an expression into simpler components called factors. Think of it as the opposite of distributing—the goal is to "un-distribute" or reverse the multiplication in an expression. For example, in the expression \(2x - 14\), we look for commonalities that we can "factor out." This often begins with identifying the greatest common factor shared among terms.

• Look for numbers or variables that evenly divide each term in the expression.
• In the expression \(2x - 14\), both terms share a common factor of 2.
• Factoring \(2x - 14\) by taking out 2, we rewrite it as \(2(x - 7)\).

Understanding this concept aids in simplifying algebraic expressions and solving equations effectively.

Greatest Common Factor

Determining the greatest common factor (GCF) is an essential step in factoring expressions. The GCF is the largest number that can divide all the terms in an expression without leaving a remainder. Finding the GCF is crucial because it helps simplify expressions and make calculations easier.

• Identify all the factors of each term in the expression.
• Choose the largest factor common to each term.
• In our expression \(2x - 14\), both terms have a factor of 2.

By factoring out the GCF, you can simplify complex expressions into more manageable parts. This step paves the way for reducing expressions to their simplest form.

Canceling Common Factors

Canceling common factors is the last step where you simplify a fraction by removing factors common to both the numerator and the denominator. It's a straightforward process once you have factored your terms correctly. Here's how to do it:

• Write the fraction in its factored form.
• Look for factors that appear in both numerator and denominator.
• Cancel them out to simplify the expression.

In our example, the fraction \(\frac{2(x - 7)}{2}\) has a common factor of 2 in both parts. By canceling out the 2, you're left with \(x - 7\), which is the simplest form. This process not only simplifies calculations but also helps in better understanding the relationships between different algebraic expressions.