Solve the following equation.Check your result in each case
`3/4(7x-1)-(2x-(1-x)/2)=x+3/2`

Answer

To solve the equation \( \frac{3}{4}(7x - 1) - \left(2x - \frac{1 - x}{2}\right) = x + \frac{3}{2} \), we will follow these steps:

### Step 1: Distribute and simplify the left-hand side
Start by distributing \( \frac{3}{4} \) to \( (7x - 1) \):

\[ \frac{3}{4} \cdot 7x - \frac{3}{4} \cdot 1 = \frac{21}{4}x - \frac{3}{4} \]

Now simplify the term \( -\left(2x - \frac{1 - x}{2}\right) \). First, simplify \( \frac{1 - x}{2} \):

\[ -\left(2x - \frac{1}{2} + \frac{x}{2}\right) = -2x + \frac{1}{2} - \frac{x}{2} = -2x + \frac{1}{2} - \frac{x}{2} = -\frac{4x}{2} - \frac{x}{2} + \frac{1}{2} = -\frac{5x}{2} + \frac{1}{2} \]

Now combine the two parts:

\[ \frac{21}{4}x - \frac{3}{4} - \frac{5x}{2} + \frac{1}{2} \]

To combine these, we need a common denominator. The common denominator for 4 and 2 is 4. Rewrite \( -\frac{5x}{2} \) and \( \frac{1}{2} \):

\[ -\frac{5x}{2} = -\frac{10x}{4}, \quad \frac{1}{2} = \frac{2}{4} \]

Now substitute back into the equation:

\[ \frac{21}{4}x - \frac{3}{4} - \frac{10}{4}x + \frac{2}{4} = \frac{21}{4}x - \frac{10}{4}x - \frac{3}{4} + \frac{2}{4} \]

Combine like terms:

\[ \frac{21 - 10}{4}x - \frac{3 - 2}{4} = \frac{11}{4}x - \frac{1}{4} \]

### Step 2: Set the equation
Now we have:

\[ \frac{11}{4}x - \frac{1}{4} = x + \frac{3}{2} \]

### Step 3: Eliminate fractions
To eliminate the fractions, multiply the entire equation by 4:

\[ 4\left(\frac{11}{4}x - \frac{1}{4}\right) = 4\left(x + \frac{3}{2}\right) \]

This simplifies to:

\[ 11x - 1 = 4x + 6 \]

### Step 4: Solve for \( x \)
Now, isolate \( x \):

\[ 11x - 4x = 6 + 1 \]
\[ 7x = 7 \]
\[ x = 1 \]

### Step 5: Check the result
Substituting \( x = 1 \) back into the original equation:

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

Calculating the left-hand side:

\[ \frac{3}{4}(7 - 1) - (2 - 0) = \frac{3}{4} \cdot 6 - 2 = \frac{18}{4} - 2 = \frac{18}{4} - \frac{8}{4} = \frac{10}{4} = \frac{5}{2} \]

Calculating the right-hand side:

\[ 1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2} \]

Since both sides are equal, the solution is verified.

### Final Answer
Thus, the solution to the equation is:

\[ \boxed{1} \]