Solve for $x$ in the equation $\frac{2}{3}x - \frac{5}{6} = \frac{3}{4}$.

1. Problem Description

Solve for $x$ in the equation $\frac{2}{3}x - \frac{5}{6} = \frac{3}{4}$.

2. Solution Steps

First, add $\frac{5}{6}$ to both sides of the equation.

$\frac{2}{3}x - \frac{5}{6} + \frac{5}{6} = \frac{3}{4} + \frac{5}{6}$

$\frac{2}{3}x = \frac{3}{4} + \frac{5}{6}$

To add the fractions, we need a common denominator. The least common multiple of 4 and 6 is

1

2. $\frac{2}{3}x = \frac{3 \cdot 3}{4 \cdot 3} + \frac{5 \cdot 2}{6 \cdot 2}$

$\frac{2}{3}x = \frac{9}{12} + \frac{10}{12}$

$\frac{2}{3}x = \frac{19}{12}$

Now, multiply both sides of the equation by $\frac{3}{2}$ to isolate $x$.

$\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot \frac{19}{12}$

$x = \frac{3 \cdot 19}{2 \cdot 12}$

$x = \frac{57}{24}$

Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is

3. $x = \frac{57 \div 3}{24 \div 3}$

$x = \frac{19}{8}$

3. Final Answer

$x = \frac{19}{8}$