The equation given in the image is:

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

Let's solve for $x$ step by step.

Step 1: Eliminate the fractions

To eliminate the fractions, find the least common denominator (LCD) for the denominators 4, 2, and 3. The LCD is 12. Multiply every term by 12 to clear the fractions:

$12 \cdot \left(\frac{3}{4}x\right) - 12 \cdot \left(\frac{1}{2}\right) = 12 \cdot \left(\frac{1}{3}x\right) + 12 \cdot 2$

This simplifies to:

$9x - 6 = 4x + 24$

Step 2: Move all terms involving $x$ to one side

Subtract $4x$ from both sides:

$9x - 4x - 6 = 24$

Simplifying further:

$5x - 6 = 24$

Step 3: Isolate $x$

Add 6 to both sides:

$5x = 30$

Finally, divide by 5:

$x = 6$

Final Answer:

$x = 6$

Would you like more details or have any other questions?

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Relative Questions:

1. What would happen if the equation had more complex fractions?
2. How can you check if the solution $x = 6$ is correct?
3. What if the equation had no solutions? How would you recognize that?
4. How would the approach change if the equation was quadratic instead of linear?
5. Can this method be applied to equations with variables on both sides involving different powers of $x$?

Tip: When solving equations with fractions, multiplying through by the least common denominator early can simplify the process significantly.