Answer:
Let's address each part of the problem:
### Part 1:
For the binary operation \(a * b = ab + a + b\), if the identity element is zero, we want to find the inverse of 2.
\[ a * b = ab + a + b \]
\[ 2 * x = 2x + 2 + x \]
Now, set this equal to the identity element (zero) and solve for \(x\):
\[ 2x + 2 + x = 0 \]
\[ 3x + 2 = 0 \]
\[ 3x = -2 \]
\[ x = -\frac{2}{3} \]
So, the inverse of 2 under this operation is \(-\frac{2}{3}\).
### Part 2:
For the binary operation \(x * y = xy - y - x\), if \(x * 3 = 2 * x\), we need to find \(x\).
\[ x * 3 = 2 * x \]
\[ (x * 3) - (2 * x) = 0 \]
\[ (3x - 3) - (2x) = 0 \]
\[ 3x - 3 - 2x = 0 \]
\[ x - 3 = 0 \]
\[ x = 3 \]
So, \(x = 3\) satisfies the equation.
### Part 3:
For the binary operation \(x * y = x + y - xy\), if \((x * 2) + (x * 3) = 68\), find \(x\).
\[ (x * 2) + (x * 3) = 68 \]
\[ (x + 2 - 2x) + (x + 3 - 3x) = 68 \]
\[ x + 2 - 2x + x + 3 - 3x = 68 \]
\[ -4x + 5 = 68 \]
\[ -4x = 63 \]
\[ x = -\frac{63}{4} \]
So, the solution for \(x\) is \(-\frac{63}{4}\).
