To solve the inequality \(-1 < 3 - 2x \leq 5\), we will break it down into two parts and solve each part step-by-step. ### Step 1: Break Down the Compound Inequality The compound inequality consists of two parts: 1. \(-1 < 3 - 2x\) 2. \(3 - 2x \leq 5\) We will solve each part separately. ### Step 2: Solve the First Inequality \(-1 < 3 - 2x\) 1. **Isolate the term with \(x\)**: \[ -1 < 3 - 2x \] Subtract 3 from both sides: \[ -1 - 3 < -2x \] This simplifies to: \[ -4 < -2x \] 2. **Divide by -2** (remember to reverse the inequality sign when dividing by a negative number): \[ \frac{-4}{-2} > x \] This simplifies to: \[ 2 > x \quad \text{or} \quad x < 2 \] ### Step 3: Solve the Second Inequality \(3 - 2x \leq 5\) 1. **Isolate the term with \(x\)**: \[ 3 - 2x \leq 5 \] Subtract 3 from both sides: \[ -2x \leq 5 - 3 \] This simplifies to: \[ -2x \leq 2 \] 2. **Divide by -2** (again, reverse the inequality sign): \[ x \geq \frac{2}{-2} \] This simplifies to: \[ x \geq -1 \] ### Step 4: Combine the Results Now we have two inequalities: 1. \(x < 2\) 2. \(x \geq -1\) Combining these gives us: \[ -1 \leq x < 2 \] ### Step 5: Determine the Integer Solutions The integer values of \(x\) that satisfy \(-1 \leq x < 2\) are: - \(-1\) - \(0\) - \(1\) ### Conclusion: Correct Option The integer values of \(x\) that satisfy the inequality are \(-1, 0, 1\). Therefore, the correct option is **D**. ### Explanation of Other Options - **Option A: -1, 0, 1, 2**: This option includes 2, which does not satisfy the inequality \(x < 2\). - **Option B: -2, 1, 0, -1**: This option includes -2, which does not satisfy the lower bound \(x \geq -1\). - **Option C: 0, 1, 2**: This option includes 2, which again does not satisfy \(x < 2\). ### Revision Summary - Solve compound inequalities by breaking them into parts. - Isolate the variable and remember to reverse the inequality sign when dividing by a negative number. - Combine the results to find the range of possible values. - Check each option against the derived solution to confirm correctness.