Simplify the expression: $\frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{3}+\frac{1}{x}}$

Question

04-11-2026
Mathematics

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Simplify the expression: $\frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{3}+\frac{1}{x}}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T07:48:27-05:00

Express the numerator and denominator as individual fractions by finding common denominators: $\frac{x^2 - 9}{3x}$ and $\frac{x + 3}{3x}$.
Divide the two fractions by multiplying the numerator by the reciprocal of the denominator: $\frac{x^2 - 9}{3x} \cdot \frac{3x}{x + 3}$.
Simplify the fraction by canceling the common term $3x$, resulting in $\frac{x^2 - 9}{x + 3}$.
Factor the numerator $x^2 - 9 = (x - 3)(x + 3)$ and cancel the term $(x + 3)$ to obtain the final simplified result: $\boxed{x - 3}$.

Explanation

Analyzing the Expression and Constraints

To simplify the expression $\frac{\frac{x}{3} - \frac{3}{x}}{\frac{1}{3} + \frac{1}{x}}$, we start by analyzing the structure of the numerator and the denominator. The numerator is $\frac{x}{3} - \frac{3}{x}$ and the denominator is $\frac{1}{3} + \frac{1}{x}$. The expression is well-defined provided that $x eq 0$ and $\frac{1}{3} + \frac{1}{x} eq 0$. Solving for the second condition, we find that $\frac{1}{x} eq -\frac{1}{3}$, so $x eq -3$. Therefore, we assume $x eq 0$ and $x eq -3$ throughout our simplification process.

Simplifying Numerator and Denominator Separately

First, we simplify the numerator, $\frac{x}{3} - \frac{3}{x}$. To subtract these fractions, we find a common denominator, which is $3x$:
\[\frac{x}{3} - \frac{3}{x} = \frac{x \cdot x}{3x} - \frac{3 \cdot 3}{3x} = \frac{x^2 - 9}{3x}\]
Next, we simplify the denominator, $\frac{1}{3} + \frac{1}{x}$. Finding a common denominator, which is $3x$, we get:
\[\frac{1}{3} + \frac{1}{x} = \frac{1 \cdot x}{3x} + \frac{1 \cdot 3}{3x} = \frac{x + 3}{3x}\]

Dividing the Simplified Fractions

Now we rewrite the original expression as the quotient of these two simplified fractions:
\[\frac{\frac{x^2 - 9}{3x}}{\frac{x + 3}{3x}}\]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[\frac{x^2 - 9}{3x} \cdot \frac{3x}{x + 3} = \frac{x^2 - 9}{x + 3}\]
Since $3x$ is in both the numerator and denominator (and $x eq 0$), we can cancel it out.

Factoring and Final Simplification

Finally, we simplify $\frac{x^2 - 9}{x + 3}$ by factoring the numerator. Recognizing that $x^2 - 9$ is a difference of squares, we have $x^2 - 9 = (x - 3)(x + 3)$. Thus, the expression becomes:
\[\frac{(x - 3)(x + 3)}{x + 3}\]
Since $x eq -3$, we can cancel the common factor $(x + 3)$ from the numerator and denominator:
\[\frac{(x - 3)(x + 3)}{x + 3} = x - 3\]
So, the simplified form of the expression is $\boxed{x - 3}$.

Examples

Simplifying complex algebraic expressions is a foundational skill in fields like engineering and physics, where complex models often need to be reduced to simpler, more manageable forms. For instance, in electrical engineering, when analyzing circuits, you might encounter complicated expressions representing equivalent impedance that need to be simplified to understand the circuit's behavior at specific frequencies. Mastering these algebraic manipulations allows for clearer analysis and more efficient problem-solving in various technical disciplines.