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

1. Problem Description

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

2. Solution Steps

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

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

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

Next, find a common denominator for the fractions on the right side. The least common multiple of 4 and 6 is

1

2. Convert the fractions to have a denominator of

1

2. $$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$$

$$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$$

So,

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

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

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

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

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

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

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

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

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

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

3. Final Answer

The final answer is $\frac{19}{8}$.