If $\frac{1}{3} + \frac{1}{2} + \frac{1}{x} = 4$, then $x =$

### Concept & Formula This is a linear equation involving fractions. To solve it efficiently, isolate the variable term on one side by moving all constant numerical fractions to the opposite side. $$ \frac{1}{x} = \text{Total Value} - (\text{Sum of known fractions}) $$ ### Step-by-Step Solution * The given algebraic equation is: $\frac{1}{3} + \frac{1}{2} + \frac{1}{x} = 4$ * First, find a common denominator for the known fractions on the left side. The LCM of $3$ and $2$ is $6$: $\frac{1}{3} = \frac{2}{6}$ $\frac{1}{2} = \frac{3}{6}$ * Add these fractions together: $\frac{2}{6} + \frac{3}{6} = \frac{5}{6}$ * Substitute this combined value back into the original equation: $\frac{5}{6} + \frac{1}{x} = 4$ * Isolate the term containing $x$ by subtracting $\frac{5}{6}$ from both sides: $\frac{1}{x} = 4 - \frac{5}{6}$ * Convert the integer $4$ into a fraction with $6$ as the denominator to perform the subtraction: $\frac{1}{x} = \frac{24}{6} - \frac{5}{6} = \frac{19}{6}$ * Finally, invert both sides to solve for the variable $x$: $x = \frac{6}{19}$ ### Exam Strategy & Shortcut Memorize common fraction sums for rapid calculation: $\frac{1}{2} + \frac{1}{3}$ is always $\frac{5}{6}$. Once you spot this, the equation immediately simplifies to $\frac{1}{x} = 4 - \frac{5}{6}$. Since $4$ is equivalently $\frac{24}{6}$, the difference is directly $\frac{19}{6}$. Flip it to get $\frac{6}{19}$. This entire sequence can be executed mentally in under 15 seconds. ### Common Pitfall Students often correctly solve for $\frac{1}{x} = \frac{19}{6}$ but accidentally stop there, assuming they have found the final answer. Always take the final crucial step to invert the result, ensuring you solve for $x$ and not $\frac{1}{x}$. ### Final Answer **Therefore, the correct answer is $\frac{6}{19}$.**