To solve the problem, we need to find the inverse of the number 2 under the binary operation defined by \( a * b = ab + a + b \). The identity element for this operation is given as zero, which means that for any number \( a \), the operation \( a * 0 \) should equal \( a \). ### Step 1: Understanding the Identity Element The identity element \( e \) for a binary operation is defined such that for any element \( a \): \[ a * e = a \] Given that the identity element is zero, we can verify this: \[ a * 0 = a \cdot 0 + a + 0 = 0 + a + 0 = a \] This confirms that zero is indeed the identity element for this operation. ### Step 2: Finding the Inverse The inverse of an element \( a \) under a binary operation is defined as an element \( b \) such that: \[ a * b = e \] In our case, we need to find \( b \) such that: \[ 2 * b = 0 \] ### Step 3: Applying the Operation Using the definition of the operation, we can substitute \( a = 2 \) and \( b \) into the equation: \[ 2 * b = 2b + 2 + b \] Setting this equal to the identity element (which is 0): \[ 2b + 2 + b = 0 \] ### Step 4: Simplifying the Equation Now, we simplify the equation: \[ 2b + b + 2 = 0 \] This simplifies to: \[ 3b + 2 = 0 \] ### Step 5: Solving for \( b \) To isolate \( b \), we subtract 2 from both sides: \[ 3b = -2 \] Now, divide both sides by 3: \[ b = -\frac{2}{3} \] ### Conclusion Thus, the inverse of 2 under the operation \( * \) is: \[ b = -\frac{2}{3} \] ### Step 6: Evaluating the Options Now, let's evaluate the options provided: - **A. \( \frac{2}{3} \)**: This is incorrect because it does not satisfy the equation \( 2 * b = 0 \). - **B. \( \frac{1}{2} \)**: This is incorrect for the same reason; it does not yield the identity element when used with 2. - **C. \( -\frac{1}{2} \)**: This is also incorrect; it does not satisfy the equation. - **D. \( -\frac{2}{3} \)**: This is correct as we derived it from the operation. ### Revision Summary - The identity element for the operation \( a * b = ab + a + b \) is 0. - To find the inverse of 2, we set up the equation \( 2 * b = 0 \). - Solving the equation \( 3b + 2 = 0 \) gives \( b = -\frac{2}{3} \). - The correct answer is option D: \( -\frac{2}{3} \).