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 equation \( a * 0 = a \) should hold true. ### Step 1: Understanding the Identity Element First, let's verify that zero is indeed the identity element for this operation. We need to check if \( a * 0 = a \): \[ a * 0 = a \cdot 0 + a + 0 = 0 + a + 0 = a \] This confirms that zero is the identity element. ### Step 2: Finding the Inverse The inverse of a number \( a \) under a binary operation is a number \( b \) such that: \[ a * b = 0 \] In our case, we want to find the inverse of 2, so we set up the equation: \[ 2 * b = 0 \] Using the definition of the operation, we can substitute: \[ 2 * b = 2b + 2 + b \] This simplifies to: \[ 2b + 2 + b = 0 \] Combining like terms gives us: \[ 3b + 2 = 0 \] ### Step 3: Solving for \( b \) Now, we can solve for \( b \): \[ 3b = -2 \] \[ b = -\frac{2}{3} \] Thus, the inverse of 2 under the operation \( * \) is \( -\frac{2}{3} \). ### Step 4: Verifying the Inverse To ensure that our solution is correct, we can verify that \( 2 * (-\frac{2}{3}) = 0 \): \[ 2 * \left(-\frac{2}{3}\right) = 2 \left(-\frac{2}{3}\right) + 2 + \left(-\frac{2}{3}\right) \] Calculating each term: 1. \( 2 \left(-\frac{2}{3}\right) = -\frac{4}{3} \) 2. \( 2 = \frac{6}{3} \) 3. \( -\frac{2}{3} \) Now, substituting these values back into the equation: \[ -\frac{4}{3} + \frac{6}{3} - \frac{2}{3} = -\frac{4}{3} + \frac{6}{3} - \frac{2}{3} = 0 \] This confirms that our calculated inverse is indeed correct. ### Step 5: Analyzing the Options Now, let's analyze the provided options: - **A. \( \frac{2}{3} \)**: This is incorrect because it does not satisfy the equation \( 2 * b = 0 \). - **B. \( \frac{1}{2} \)**: This is also incorrect for the same reason; it does not yield zero when substituted into the operation. - **C. \( -\frac{1}{2} \)**: This option is incorrect as well; substituting it does not satisfy the inverse condition. - **D. \( -\frac{2}{3} \)**: This is the correct answer, as we have shown that it satisfies \( 2 * (-\frac{2}{3}) = 0 \). ### Summary - The binary operation is defined as \( a * b = ab + a + b \). - The identity element is zero, confirmed by \( a * 0 = a \). - The inverse of 2 is found by solving \( 2 * b = 0 \), leading to \( b = -\frac{2}{3} \). - Verification shows that \( 2 * (-\frac{2}{3}) = 0 \), confirming the correctness of the inverse. ### Revision Summary - The operation \( a * b = ab + a + b \) has an identity element of 0. - To find the inverse, set \( a * b = 0 \) and solve for \( b \). - The inverse of 2 is \( -\frac{2}{3} \). - Always verify your solution by substituting back into the operation.