Find $x$ if $\frac{x}{5} - \frac{x}{6} = 4$.

### Concept & Formula Solving this equation requires isolating the variable $x$. Since $x$ appears in fractions with different denominators, the primary algebraic strategy is to find a common denominator, combine the fractions into a single term, and then solve for $x$. $$ \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} $$ ### Step-by-Step Solution * **Given:** * Equation: $\frac{x}{5} - \frac{x}{6} = 4$ * **Calculation:** 1. Find the Least Common Multiple (LCM) for the denominators $5$ and $6$. The LCM is $30$. 2. Rewrite both fractions so they share this common denominator. * Multiply the first fraction by $\frac{6}{6}$: $\frac{6x}{30}$. * Multiply the second fraction by $\frac{5}{5}$: $\frac{5x}{30}$. 3. Substitute these back into the equation: $\frac{6x}{30} - \frac{5x}{30} = 4$. 4. Combine the numerators over the single denominator: $\frac{6x - 5x}{30} = 4$. 5. Simplify the numerator: $\frac{x}{30} = 4$. 6. Multiply both sides of the equation by $30$ to isolate $x$. 7. $x = 4 \cdot 30 = 120$. ### Exam Strategy & Shortcut An alternative to finding common denominators is to multiply the *entire* equation by the LCM ($30$) immediately to clear all fractions in one step. $30 \cdot (\frac{x}{5}) - 30 \cdot (\frac{x}{6}) = 30 \cdot (4)$ $6x - 5x = 120$ $x = 120$. This method is less prone to error and gets you to the answer much faster on exams. ### Common Pitfall The most frequent error is forgetting to multiply the constant on the right side of the equals sign ($4$) when clearing denominators. Students will often write $6x - 5x = 4 \implies x = 4$. Always remember that whatever operation you do to one side of an equation, you must do to every term on both sides. ### Final Answer Therefore, the correct answer is 120.