Problem 44 - Elementary and Intermediate Algebra

Question

Simplify. See Example 4. $$ \frac{4 x+12}{16} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x+3}{4}\).

Step 1: Identify Common Factors

First, identify the greatest common factor (GCF) of the terms in the numerator. In this case, the terms in the numerator are \(4x\) and \(12\). Both terms are divisible by \(4\), which is the GCF.

Step 2: Factor the Numerator

Factor \(4\) out of the numerator. This means writing the numerator \(4x + 12\) as \(4(x + 3)\) by dividing each term by \(4\).

Step 3: Simplify the Fraction

Rewrite the fraction using the factored numerator: \(\frac{4(x+3)}{16}\). Next, simplify by dividing both the numerator and the denominator by their common factor \(4\). The result is \(\frac{x+3}{4}\).

Step 4: Final Simplified Expression

After simplifying, the fraction has been reduced to its simplest form: \(\frac{x+3}{4}\). Therefore, the simplified expression is \(\frac{x+3}{4}\).

Key Concepts

Simplification

Simplification in algebra involves reducing an expression to its simplest form. This makes equations easier to work with. Simplifying algebraic fractions means reducing the fraction by removing common factors in the numerator and denominator.

For example, in the exercise of simplifying \( \frac{4x + 12}{16} \), an essential first step is to simplify the structure of the expression. By simplifying, you'll develop a more manageable equation that retains the same value as the original. It involves knowing the properties of numbers and operations.

By simplifying a fraction, you help to better see and understand the relationships between the components of an expression. Always start by looking for common factors of both numerator and denominator, simplifying the fraction step by step.

Greatest Common Factor

The greatest common factor (GCF) is a crucial concept when working with fractions. The GCF of two or more numbers is the largest number that divides all of them without leaving a remainder.

Finding the GCF:

• Identify all divisors of the numbers in question.
• Choose the largest divisor common to all numbers involved.
• In the example, the numbers in the numerator are \(4x\) and \(12\). Both are divisible by \(4\), so \(4\) is the GCF.

Using the GCF simplifies the process of reducing fractions. By dividing each term of the expression by this factor, you are reducing it to its simplest form while maintaining the equation's balance and equality.

Factoring

Factoring is the process of breaking down numbers or expressions into their constituent factors or numbers that multiply together to give the original number or expression.

When you factor the numerator in a fraction, you simplify it by expressing it in terms of its basic building blocks. This is especially useful for simplification of algebraic fractions.

To factor effectively:

• Find the GCF of all terms in the numerator.
• Divide each term by the GCF, rewriting the expression as a product of the GCF and the simplified terms.
• In the given problem, \(4x + 12\) can be rewritten as \(4(x + 3)\) by factoring out the \(4\).

Factoring is a fundamental skill in simplifying expressions, as it allows you to see the structure and potential simplifications easily. It reduces complex expressions to more manageable forms and is a stepping stone to solving various algebraic problems.