Consider the binary operation '*' on `R-{-2}` defined a * b `=a+b+(ab)/(2)`, for all `a,b in R-{-2}`. Find the value of x if 1 * (x * 2) = 8.

Answer

To solve the problem, we need to find the value of \( x \) such that \( 1 * (x * 2) = 8 \), given the binary operation defined as \( a * b = a + b + \frac{ab}{2} \).

### Step-by-Step Solution:

1. **Calculate \( x * 2 \)**:
Using the operation defined, we can express \( x * 2 \):
\[ x * 2 = x + 2 + \frac{x \cdot 2}{2} \]
Simplifying this, we get:
\[ x * 2 = x + 2 + x = 2x + 2 \]

2. **Substitute \( x * 2 \) into \( 1 * (x * 2) \)**:
Now we substitute \( x * 2 \) into the expression \( 1 * (x * 2) \):
\[ 1 * (x * 2) = 1 * (2x + 2) \]
Using the operation again:
\[ 1 * (2x + 2) = 1 + (2x + 2) + \frac{1 \cdot (2x + 2)}{2} \]
Simplifying this:
\[ 1 * (2x + 2) = 1 + 2x + 2 + \frac{2x + 2}{2} \]
\[ = 1 + 2x + 2 + (x + 1) = 2x + x + 4 = 3x + 4 \]

3. **Set the equation equal to 8**:
We are given that \( 1 * (x * 2) = 8 \):
\[ 3x + 4 = 8 \]

4. **Solve for \( x \)**:
Subtract 4 from both sides:
\[ 3x = 8 - 4 \]
\[ 3x = 4 \]
Now, divide by 3:
\[ x = \frac{4}{3} \]

### Final Answer:
The value of \( x \) is \( \frac{4}{3} \).