Problem 100 - Introductory Algebra for College Students

Question

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. The rational expressions $$\frac{7}{14 x} and \frac{7}{14+x}$$ an both be simplified by dividing each numerator and each denominator by 7.

Step-by-Step Solution

Answer

The given statement only makes sense for the first rational expression \( \frac{7}{14x} \), but does not make sense for the second rational expression \( \frac{7}{14+x} \).

Step 1: Simplify first expression

If we divide both the numerator and the denominator of the first expression \( \frac{7}{14x} \) by 7, we get \( \frac{1}{2x} \). So, the given statement is correct for the first expression.

Step 2: Simplify second expression

If we divide both the numerator and the denominator of the second expression \( \frac{7}{14+x} \) by 7, we get \( \frac{1}{2+\frac{x}{7}} \). Now, if we multiply both the numerator and the denominator by 7 to eliminate the fraction in the denominator, we finally get \( \frac{7}{14+x} \), which is the original expression. So, the given statement is not correct for the second expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is crucial in algebra to make complex equations more manageable. When you simplify an expression, you aim to make it easier by finding its simplest form without changing its value. For rational expressions like fractions, we often simplify by dividing the numerator and the denominator by their greatest common divisor (GCD).
This process involves identifying what each part of the fraction can be equally divided by without leaving a remainder. Simplifying helps in solving and understanding various mathematical problems more clearly. However, care must be taken to apply this correctly to avoid incorrect results.

Numerator and Denominator

In any rational expression, you will find two key components: the numerator and the denominator.

• The numerator is the top number or expression, indicating how many parts of the whole are being considered.
• The denominator is the bottom number, which represents the total number of equal parts the whole is divided into.

Understanding these terms is essential when performing operations such as addition, subtraction, or simplification of fractions. Each operation involves manipulating these two components under specific mathematical rules to maintain the expression's balance and equivalence.

Incorrect Simplification

Incorrect simplification of rational expressions can lead to errors that change the expression's meaning or value. A common mistake is dividing terms inside a sum or difference incorrectly. For example, in the expression \( \frac{7}{14+x} \), it might be tempting to divide the entire denominator by 7. However, this isn't correct because the division applies only when the terms are factors of a whole multiplied product, not when they are mixed with addition or subtraction.
Using care with these operations ensures you don't inadvertently alter the expression while attempting to simplify.

Algebraic Manipulation

Algebraic manipulation involves rearranging and modifying algebraic expressions to achieve a specific goal. This could be simplifying an expression, solving an equation, or rewriting it in a more convenient form. Using algebraic manipulation, one can help uncover the underlying structure of a mathematical problem. In rational expressions, manipulation often involves factoring, expanding, adding, or simplifying terms to manage the complexities of algebraic calculations. While powerful, manipulation requires a proper understanding of mathematical laws and properties to ensure changes made are valid and do not alter the expression's original value.
By mastering these techniques, one enhances their ability to solve various mathematical problems effectively.