To solve the inequality \(2 - x > x^2\), we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Rearranging the Inequality First, we want to rearrange the inequality so that all terms are on one side. We can do this by moving \(x^2\) and \(x\) to the left side: \[ 2 - x - x^2 > 0 \] This can be rewritten as: \[ -x^2 - x + 2 > 0 \] ### Step 2: Reorganizing the Terms To make it easier to factor, we can multiply the entire inequality by -1 (remember that this reverses the inequality sign): \[ x^2 + x - 2 < 0 \] ### Step 3: Factoring the Quadratic Next, we need to factor the quadratic expression \(x^2 + x - 2\). We are looking for two numbers that multiply to -2 (the constant term) and add to 1 (the coefficient of \(x\)). The numbers that satisfy this are 2 and -1. Thus, we can factor the quadratic as follows: \[ (x + 2)(x - 1) < 0 \] ### Step 4: Finding the Critical Points The critical points occur when the expression equals zero: \[ (x + 2) = 0 \quad \Rightarrow \quad x = -2 \] \[ (x - 1) = 0 \quad \Rightarrow \quad x = 1 \] These critical points divide the number line into intervals. We will test each interval to determine where the inequality holds true. ### Step 5: Testing the Intervals The critical points divide the number line into three intervals: 1. \( (-\infty, -2) \) 2. \( (-2, 1) \) 3. \( (1, \infty) \) We will test a point from each interval in the inequality \( (x + 2)(x - 1) < 0 \). - **Interval 1: \( (-\infty, -2) \)** Choose \(x = -3\): \[ (-3 + 2)(-3 - 1) = (-1)(-4) = 4 \quad (\text{not less than } 0) \] - **Interval 2: \( (-2, 1) \)** Choose \(x = 0\): \[ (0 + 2)(0 - 1) = (2)(-1) = -2 \quad (\text{less than } 0) \] - **Interval 3: \( (1, \infty) \)** Choose \(x = 2\): \[ (2 + 2)(2 - 1) = (4)(1) = 4 \quad (\text{not less than } 0) \] ### Step 6: Conclusion The inequality \( (x + 2)(x - 1) < 0 \) holds true in the interval \( (-2, 1) \). Therefore, the solution to the inequality \( 2 - x > x^2 \) is: \[ -2 < x < 1 \] ### Final Answer The correct option is **D. -2 < x < 1**. ### Explanation of Other Options - **Option A: \(x < -2\) or \(x > 1\)** This option suggests that the inequality holds outside the interval we found, which is incorrect based on our testing. - **Option B: \(x > 2\) or \(x < -1\)** This option incorrectly includes values greater than 2, which do not satisfy the inequality. - **Option C: \(-1 < x < 2\)** This option incorrectly includes values greater than 1, which do not satisfy the inequality. ### Revision Summary - Rearranged the inequality to \(x^2 + x - 2 < 0\). - Factored the quadratic to \((x + 2)(x - 1) < 0\). - Tested intervals to find where the inequality holds. - Concluded that the solution is \(-2 < x < 1\).