To find the inverse of \( p \) under the binary operation defined by \( p * q = p + q - pq \), we need to determine a value \( x \) such that \( p * x = 0 \), where 0 is the identity element for this operation. ### Step-by-Step Explanation 1. **Understanding the Operation**: The operation \( * \) is defined as: \[ p * q = p + q - pq \] This means that when we combine \( p \) and \( q \) using this operation, we add them together and then subtract the product of \( p \) and \( q \). 2. **Setting Up the Equation**: We want to find \( x \) such that: \[ p * x = 0 \] Substituting the operation into this equation gives: \[ p + x - px = 0 \] 3. **Rearranging the Equation**: To isolate \( x \), we can rearrange the equation: \[ p + x - px = 0 \implies x - px = -p \] Factoring out \( x \) from the left side: \[ x(1 - p) = -p \] 4. **Solving for \( x \)**: Now, we can solve for \( x \): \[ x = \frac{-p}{1 - p} \] To make this expression more manageable, we can multiply the numerator and denominator by -1: \[ x = \frac{p}{p - 1} \] 5. **Identifying the Inverse**: Thus, the inverse of \( p \) under the operation \( * \) is: \[ x = \frac{p}{p - 1} \] ### Evaluating the Options Now, let's compare our derived inverse \( \frac{p}{p - 1} \) with the provided options: - **Option A: \( p \)** - This is incorrect because the identity element is 0, and \( p * p \) does not equal 0 for all \( p \). - **Option B: \( p - 1 \)** - This is incorrect because substituting \( p - 1 \) into the operation does not yield the identity element 0. - **Option C: \( \frac{p}{p - 1} \)** - This is correct as we derived this expression for the inverse of \( p \). - **Option D: \( \frac{p}{p + 1} \)** - This is incorrect because it does not match our derived expression and does not satisfy the condition \( p * x = 0 \). ### Summary of Key Points - The operation defined is \( p * q = p + q - pq \). - To find the inverse of \( p \), we set up the equation \( p * x = 0 \) and solved for \( x \). - The correct inverse is \( \frac{p}{p - 1} \), which corresponds to option C. - Other options do not satisfy the condition for being the inverse under the defined operation. ### Revision Summary - The operation \( p * q = p + q - pq \) has an identity element of 0. - The inverse of \( p \) is found by solving \( p * x = 0 \). - The derived inverse is \( \frac{p}{p - 1} \). - Always check each option against the derived expression to confirm correctness.