The equation displayed is:

$\frac{3(7x - 1)}{4} - \left( 2x - \frac{1 - x}{2} \right) = x + \frac{3}{2}$

Let's solve this step-by-step.

Step 1: Simplify the left-hand side

We first distribute the terms and simplify the equation on the left-hand side.

$\frac{3(7x - 1)}{4} = \frac{21x - 3}{4}$ $\left( 2x - \frac{1 - x}{2} \right) = 2x - \frac{1}{2} + \frac{x}{2}$ So the equation becomes: $\frac{21x - 3}{4} - \left(2x - \frac{1}{2} + \frac{x}{2}\right) = x + \frac{3}{2}$

Step 2: Combine like terms on the left-hand side

Now let's combine the terms in the parentheses: $2x + \frac{x}{2} - \frac{1}{2} = \frac{4x + x}{2} - \frac{1}{2} = \frac{5x - 1}{2}$

Now the equation becomes: $\frac{21x - 3}{4} - \frac{5x - 1}{2} = x + \frac{3}{2}$

Step 3: Eliminate fractions by multiplying through by the least common denominator (LCD)

The LCD of 4 and 2 is 4, so multiply every term by 4: $21x - 3 - 2(5x - 1) = 4\left(x + \frac{3}{2}\right)$

Distribute on both sides: $21x - 3 - 10x + 2 = 4x + 6$

Step 4: Combine like terms

On the left-hand side: $21x - 10x - 3 + 2 = 11x - 1$ On the right-hand side: $4x + 6$

So the equation simplifies to: $11x - 1 = 4x + 6$

Step 5: Solve for $x$

Move all terms involving $x$ to one side: $11x - 4x = 6 + 1$ $7x = 7$ $x = 1$

Final Answer:

The solution to the equation is $x = 1$.

Would you like more details on any step or further clarification?

Here are 5 related questions to expand your understanding:

1. How do you simplify complex fractions in an equation?
2. What is the process of finding the least common denominator (LCD)?
3. How do you check if a solution satisfies the original equation?
4. What are the benefits of combining like terms before solving for the variable?
5. Can similar methods be used to solve systems of equations?

Tip: Always double-check your work by substituting the solution back into the original equation to verify correctness.