Solve the equations and check your answer in the 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: Simplify the equation
Start by distributing and simplifying both sides of the equation.

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

Distributing \( \frac{3}{4} \):

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

This simplifies to:

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

### Step 2: Combine like terms
Combine the terms on the left side.

First, convert all terms to have a common denominator of 4:

\[ \frac{21x}{4} - \frac{3}{4} - \frac{8x}{4} + \frac{2}{4} - \frac{2x}{4} = x + \frac{3}{2} \]

Now combine the \( x \) terms:

\[ \frac{21x - 8x - 2x}{4} - \frac{3 - 2}{4} = x + \frac{3}{2} \]

This simplifies to:

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

### Step 3: Eliminate the fraction
Multiply the entire equation by 4 to eliminate the fraction:

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

### Step 4: Solve for \( x \)
Rearranging the equation gives:

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

This simplifies to:

\[ 7x = 7 \]

Dividing both sides by 7 gives:

\[ x = 1 \]

### Step 5: Check the solution
Substitute \( x = 1 \) back into the original equation to verify:

Left-hand side:

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

Right-hand side:

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

Since both sides are equal, our solution is verified.

### Final Answer:
The value of \( x \) is \( 1 \).

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