Solve the following inequality. \[ (x-1)(x+4)(x-5) \leq 0 \]

To solve the inequality \((x-1)(x+4)(x-5) \leq 0\), we first identify the roots of the corresponding equation \((x-1)(x+4)(x-5) = 0\). The roots are: - \(x = 1\) - \(x = -4\) - \(x = 5\) Next, we determine the intervals defined by these roots: \((- \infty, -4)\), \((-4, 1)\), \((1, 5)\), and \((5, +\infty)\). We will test a point from each interval to see where the product is less than or equal to zero. 1. **Interval \((- \infty, -4)\)**: Choose \(x = -5\) \[ (-5-1)(-5+4)(-5-5) = (-6)(-1)(-10) = -60 \quad (\text{negative}) \] 2. **Interval \((-4, 1)\)**: Choose \(x = 0\) \[ (0-1)(0+4)(0-5) = (-1)(4)(-5) = 20 \quad (\text{positive}) \] 3. **Interval \((1, 5)\)**: Choose \(x = 3\) \[ (3-1)(3+4)(3-5) = (2)(7)(-2) = -28 \quad (\text{negative}) \] 4. **Interval \((5, +\infty)\)**: Choose \(x = 6\) \[ (6-1)(6+4)(6-5) = (5)(10)(1) = 50 \quad (\text{positive}) \] Now we summarize the result: - In the interval \((- \infty, -4)\), the product is negative. - In the interval \((-4, 1)\), the product is positive. - In the interval \((1, 5)\), the product is negative. - In the interval \((5, +\infty)\), the product is positive. Next, we need to consider the equality part of the inequality. The points where the expression equals zero are \(x = -4\), \(x = 1\), and \(x = 5\). These points contribute to the solution of the inequality. Thus, the solution to the inequality \((x-1)(x+4)(x-5) \leq 0\) is: \[ (-\infty, -4] \cup [1, 5] \]