To solve the problem, we need to find the inverse of the element 2 under the operation defined by \( a * b = a + b + 1 \). We also know that the identity element for this operation is -1. Let's break this down step-by-step. ### Step 1: Understanding the Operation and Identity Element The operation \( a * b \) is defined as: \[ a * b = a + b + 1 \] The identity element is a special number \( e \) such that for any number \( a \): \[ a * e = a \] In our case, we are given that the identity element is -1. Therefore, we can check this: \[ a * (-1) = a + (-1) + 1 = a + 0 = a \] This confirms that -1 is indeed the identity element for this operation. ### Step 2: Finding the Inverse The inverse of an element \( a \) under the operation \( * \) is a number \( b \) such that: \[ a * b = e \] where \( e \) is the identity element (-1 in this case). For our specific case, we want to find the inverse of 2. Thus, we need to solve: \[ 2 * b = -1 \] ### Step 3: Applying the Operation Using the definition of the operation, we can substitute: \[ 2 * b = 2 + b + 1 \] Setting this equal to the identity element: \[ 2 + b + 1 = -1 \] ### Step 4: Solving for \( b \) Now, we simplify the equation: \[ 2 + b + 1 = -1 \] Combine like terms: \[ b + 3 = -1 \] Now, isolate \( b \): \[ b = -1 - 3 \] \[ b = -4 \] ### Conclusion: The Inverse of 2 Thus, the inverse of the element 2 under the operation \( * \) is: \[ \boxed{-4} \] ### Step 5: Analyzing the Options Now, let's look at the options provided: - **A. 4**: This is incorrect because if we substitute 4 for \( b \), we would get \( 2 * 4 = 2 + 4 + 1 = 7 \), which is not equal to -1. - **B. zero**: This is incorrect because substituting 0 for \( b \) gives \( 2 * 0 = 2 + 0 + 1 = 3 \), which is not equal to -1. - **C. -2**: This is incorrect because substituting -2 for \( b \) gives \( 2 * (-2) = 2 - 2 + 1 = 1 \), which is not equal to -1. - **D. -4**: This is correct as we calculated that the inverse of 2 is indeed -4. ### Revision Summary - The operation defined is \( a * b = a + b + 1 \). - The identity element for this operation is -1. - The inverse of an element \( a \) is found by solving \( a * b = e \). - For the element 2, the inverse is -4, as confirmed by our calculations. This thorough breakdown should help you understand how to find inverses under a defined operation and clarify why the other options do not work.