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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?

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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.

Solve the equation 3(7x - 1) / 4 - (2x - (1 - x)/2) = x + 3/2.

Learn how to solve a linear equation involving fractions and the distributive property step-by-step. This example covers key algebraic concepts such as simplifying expressions, eliminating fractions, and solving for variables. Suitable for students in Grades 8-10.

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