Solve the following inequality. \[ (x-2)(x-7)^{2}<0 \]

To solve the inequality \((x-2)(x-7)^{2}<0\), we first identify the critical points by setting the expression equal to zero. The critical points are \(x = 2\) and \(x = 7\). Next, we will analyze the sign of the expression in the intervals determined by these critical points: 1. **Interval \((- \infty, 2)\)**: Choose a test point, say \(x = 0\): \[(0-2)(0-7)^{2} = (-2)(49) < 0 \quad \text{(True)}\] 2. **Interval \((2, 7)\)**: Choose a test point, say \(x = 5\): \[(5-2)(5-7)^{2} = (3)(4) > 0 \quad \text{(False)}\] 3. **Interval \((7, \infty)\)**: Choose a test point, say \(x = 8\): \[(8-2)(8-7)^{2} = (6)(1) > 0 \quad \text{(False)}\] From this analysis, we conclude that the inequality holds true in the interval \((- \infty, 2)\). Next, we should consider the points where the expression is zero: at \(x = 2\) the expression equals zero and at \(x = 7\) it is also zero but since we are looking for where it is **less than** zero, these points will not be included in the solution. Therefore, the final answer for the inequality \((x-2)(x-7)^{2}<0\) is: \[ (-\infty, 2) \]