Which inequality has solutions that include 12,13 , and \( 14 ? \) (1 poinf) \( 02 x \leq 12 \) \( 02 x \geq 11+13 \) \( 02 x<13+15 \) \( 2 x \geq 34-6 \)

\[ \textbf{Step 1: Analyze each answer option.} \] \[ \textbf{Option 1: } 2x \leq 12 \] For \(x = 12\), we have \(2(12) = 24\), which is not \(\leq 12\). Thus, this inequality does not include 12, so it is not valid. \[ \textbf{Option 2: } 2x \geq 11+13 \] First, simplify the right-hand side: \[ 11 + 13 = 24 \] Thus, the inequality becomes: \[ 2x \geq 24 \] Dividing both sides by 2: \[ x \geq 12 \] For \(x = 12, 13, 14\): \[ x = 12 \quad \Rightarrow \quad 12 \geq 12 \quad (\text{True}) \] \[ x = 13 \quad \Rightarrow \quad 13 \geq 12 \quad (\text{True}) \] \[ x = 14 \quad \Rightarrow \quad 14 \geq 12 \quad (\text{True}) \] All required values satisfy the inequality. \[ \textbf{Option 3: } 2x < 13+15 \] Simplify the right-hand side: \[ 13 + 15 = 28 \] Thus, the inequality becomes: \[ 2x < 28 \] Divide both sides by 2: \[ x < 14 \] For \(x = 14\): \[ 14 < 14 \quad (\text{False}) \] So, \(x = 14\) does not satisfy the inequality. \[ \textbf{Option 4: } 2x \geq 34-6 \] Simplify the right-hand side: \[ 34 - 6 = 28 \] Thus, the inequality becomes: \[ 2x \geq 28 \] Divide both sides by 2: \[ x \geq 14 \] For \(x = 12\) and \(x = 13\): \[ 12 \geq 14 \quad (\text{False}) \] \[ 13 \geq 14 \quad (\text{False}) \] Thus, the inequality does not include \(x = 12\) and \(x = 13\). \[ \textbf{Step 2: Conclusion.} \] Only Option 2 (\(2x \geq 11+13\)) gives solutions that include \(12\), \(13\), and \(14\). \[ \boxed{2x \geq 11+13} \]