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The equation you provided is:

\[
x + \frac{1}{2} = \frac{4x + 3}{4} - \frac{2 - 3x}{8}
\]

### Step 1: Eliminate fractions by finding the least common denominator (LCD)
The denominators are 2, 4, and 8. The least common denominator is 8. Multiply every term in the equation by 8 to eliminate the fractions:

\[
8 \left(x + \frac{1}{2}\right) = 8 \left(\frac{4x + 3}{4}\right) - 8 \left(\frac{2 - 3x}{8}\right)
\]

This simplifies to:

\[
8x + 4 = 2(4x + 3) - (2 - 3x)
\]

### Step 2: Simplify both sides
Distribute on the right-hand side:

\[
8x + 4 = 8x + 6 - (2 - 3x)
\]

Now distribute the negative sign:

\[
8x + 4 = 8x + 6 - 2 + 3x
\]

### Step 3: Combine like terms
Simplify further by combining like terms:

\[
8x + 4 = 8x + 3x + 6 - 2
\]
\[
8x + 4 = 11x + 4
\]

### Step 4: Isolate the variable
Subtract \(8x\) from both sides:

\[
4 = 3x + 4
\]

Now subtract 4 from both sides:

\[
0 = 3x
\]

### Step 5: Solve for \(x\)
Divide both sides by 3:

\[
x = 0
\]

So, the solution to the equation is \(x = 0\).

---

Would you like any further details or explanations?

Here are some related questions you might find helpful:
1. How do you find the least common denominator in an equation?
2. Why do we multiply by the least common denominator in equations?
3. What are the properties of solving linear equations?
4. Can you explain the distributive property in this context?
5. How do you check if a solution is correct by substituting it back into the original equation?

**Tip:** Always double-check your solution by substituting it back into the original equation to ensure it satisfies the equation.

Solve: x + 2 * (1/2) = (4x + 3) / 4 - (2 - 3x) / 8

Detailed step-by-step solution to solve a linear equation involving fractions: x + 2(1/2) = (4x + 3) / 4 - (2 - 3x) / 8. Learn about eliminating fractions, using the distributive property, and combining like terms. Suitable for high school students (Grades 9-12).

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