Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ \frac{(2 x+1)(x+3)}{(2 x-1)(x+3)}=\frac{2 x+1}{2 x-1} $$

Step-by-Step Solution

Answer

The real number property that justifies the given statement is the Cancellation Law for Fractions. This is applied by cancelling out the common factor \((x+3)\) in the numerator and the denominator, resulting in \(\frac{2x+1}{2x-1}\).

Step 1: Identify common factors in the numerator and denominator

In the given statement, we can see that the numerator is the product of two expressions, \((2x+1)\) and \((x+3)\). Also, the denominator is the product of two expressions, \((2x-1)\) and \((x+3)\). Notice that both the numerator and the denominator have the factor \((x+3)\). $$ \frac{(2 x+1)(x+3)}{(2 x-1)(x+3)}=\frac{2 x+1}{2 x-1} $$

Step 2: Apply the simplification process

Now, using the simplification process, we can cancel out the common factor \((x+3)\) from the numerator and the denominator of the fraction: $$ \frac{(2 x+1)(x+3)}{(2 x-1)(x+3)}=\frac{(2x+1)\cancel{(x+3)}}{(2x-1)\cancel{(x+3)}}= \frac{2x+1}{2x-1} $$

Step 3: State the real number property

Based on the above simplification, the fraction is simplified by applying the cancellation of common factors in the numerator and the denominator. The property that justifies this statement is the Cancellation Law for Fractions.

Key Concepts

Cancellation Law for Fractions

The Cancellation Law for Fractions is a handy principle when working with algebraic fractions. It states that if both the numerator and the denominator of a fraction contain a common factor, you can "cancel" this factor out. This simplifies the fraction without changing its value.
For example, let's consider the fraction \( \frac{(2x+1)(x+3)}{(2x-1)(x+3)} \). Here, the common factor is \((x+3)\). By canceling out \((x+3)\) from both the numerator and the denominator, we get the simplified fraction \( \frac{2x+1}{2x-1} \).
However, be cautious! You can only cancel factors, not terms separated by plus or minus signs. Always factorize each part of the fraction completely to see all common factors clearly.

Simplification Process

The simplification process in algebra involves reducing expressions to their simplest form. This often involves canceling out common factors, which makes equations or expressions easier to work with and understand. When simplifying fractions like \( \frac{(2x+1)(x+3)}{(2x-1)(x+3)} \), our goal is to make the expression as simple as possible.
Here's how you do it:

• Start by factoring both the numerator and the denominator.
• Look for any shared or common factors in these factored forms.
• Cancel the common factors that appear in both the numerator and the denominator.

Thus, by applying these steps effectively, a complicated fraction could be pared down to its core elements, ready for use in equations or comparisons.

Common Factors

Understanding common factors is crucial in simplifying both numerical and algebraic fractions. Common factors are numbers or expressions that divide exactly into two or more numbers or expressions.
In our example \( \frac{(2x+1)(x+3)}{(2x-1)(x+3)} \), the common factor is \((x+3)\). This expression appears in both the numerator and the denominator, enabling us to cancel it out using the Cancellation Law.
Identifying common factors:

• First, break down the expressions into their individual factors.
• List out all factors for both the numerator and denominator.
• Spot any similar factors that appear in both lists.

When simplification occurs, it leads us directly to more straightforward calculations, highlighting the importance of spotting common factors efficiently.