Final answer:
To solve \(\frac{x}{2} + \frac{x}{6} = \frac{x}{4} + 1\), we find a common denominator and simplify the equation step by step. The solution is x = \(\frac{12}{5}\), which is a simplified fraction.
Step-by-step explanation:
To find x in the equation \(\frac{x}{2} + \frac{x}{6} = \frac{x}{4} + 1\), we want to combine all the x terms on one side and constants on the other. Our initial goal is to find a common denominator for the fractions.
We notice that 12 is a common multiple of 2, 6, and 4. Multiplying every term by 12 to clear the denominators gives us:
\(6x + 2x = 3x + 12\).
Combine like terms:
\(8x = 3x + 12\)
Subtract 3x from both sides:
\(5x = 12\)
Now divide both sides by 5 to isolate x:
\(x = \frac{12}{5}\)
Thus, x equals \(\frac{12}{5}\) which is a simplified fraction.