(x+5)*(x-1)*(x-4) <0

To solve the inequality \((x + 5)(x - 1)(x - 4) < 0\), we first need to determine the critical points, which are the values of \(x\) that make each factor equal to zero. 1. **Find the critical points:** - \(x + 5 = 0 \Rightarrow x = -5\) - \(x - 1 = 0 \Rightarrow x = 1\) - \(x - 4 = 0 \Rightarrow x = 4\) So, the critical points are \(x = -5\), \(x = 1\), and \(x = 4\). 2. **Determine the intervals:** The critical points divide the number line into four intervals: - \((-\infty, -5)\) - \((-5, 1)\) - \((1, 4)\) - \((4, +\infty)\) 3. **Test each interval:** Choose a test point from each interval and evaluate the expression \((x + 5)(x - 1)(x - 4)\): - **Interval \((-\infty, -5)\):** Choose \(x = -6\) \[ (-6 + 5)(-6 - 1)(-6 - 4) = (-1)(-7)(-10) = -70 \quad (\text{negative}) \] - **Interval \((-5, 1)\):** Choose \(x = 0\) \[ (0 + 5)(0 - 1)(0 - 4) = (5)(-1)(-4) = 20 \quad (\text{positive}) \] - **Interval \((1, 4)\):** Choose \(x = 2\) \[ (2 + 5)(2 - 1)(2 - 4) = (7)(1)(-2) = -14 \quad (\text{negative}) \] - **Interval \((4, +\infty)\):** Choose \(x = 5\) \[ (5 + 5)(5 - 1)(5 - 4) = (10)(4)(1) = 40 \quad (\text{positive}) \] 4. **Summary of test results:** - In \((-\infty, -5)\), the expression is negative. - In \((-5, 1)\), the expression is positive. - In \((1, 4)\), the expression is negative. - In \((4, +\infty)\), the expression is positive. 5. **Determine where the expression is negative:** From our test results, \((x + 5)(x - 1)(x - 4) < 0\) in the intervals: - \((-\infty, -5)\) - \((1, 4)\) Thus, the solution to the inequality \((x + 5)(x - 1)(x - 4) < 0\) is: \[ x \in (-\infty, -5) \cup (1, 4) \]