Solving for x in Algebraic Equation

The problem requires finding the value of '$x$' from the given equation:

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

Equation Rearrangement

To solve for '$x$', we first group the terms containing '$x$' on one side and the constant terms on the other side of the equation.

$ \frac{1}{x} - \frac{1}{2x} = \frac{3}{4} - \frac{2}{3} $

Combining Terms

Next, we find a common denominator for the terms on each side to combine them.

• Left side: The common denominator for '$x$' and '$2x$' is '$2x$'. $ \frac{2}{2x} - \frac{1}{2x} = \frac{2 - 1}{2x} = \frac{1}{2x} $
• Right side: The common denominator for '4' and '3' is '12'. $ \frac{3 \times 3}{4 \times 3} - \frac{2 \times 4}{3 \times 4} = \frac{9}{12} - \frac{8}{12} = \frac{9 - 8}{12} = \frac{1}{12} $

Finding x Value

Now, the equation simplifies to:

$ \frac{1}{2x} = \frac{1}{12} $

Since the numerators are equal, the denominators must also be equal.

$ 2x = 12 $

Finally, solve for '$x$' by dividing both sides by 2:

$ x = \frac{12}{2} $

$ x = 6 $