(x+1) (x-2) (x-4) <0
To solve the inequality \((x+1)(x-2)(x-4) < 0\), follow these steps:
Find the roots: Set the expression equal to zero to find the critical points: \[ (x+1)(x-2)(x-4) = 0 \] The roots are \(x = -1\), \(x = 2\), and \(x = 4\).
Determine the intervals: The critical points divide the number line into the following intervals:
- \((-\infty, -1)\)
- \((-1, 2)\)
- \((2, 4)\)
- \((4, \infty)\)
Test each interval: Choose a test point from each interval to determine the sign of the product in that interval.
Interval \((-\infty, -1)\) (test point \(x = -2\)): \[ (-2 + 1)(-2 - 2)(-2 - 4) = (-1)(-4)(-6) = -24 < 0 \] (This interval satisfies the inequality.)
Interval \((-1, 2)\) (test point \(x = 0\)): \[ (0 + 1)(0 - 2)(0 - 4) = (1)(-2)(-4) = 8 > 0 \] (This interval does not satisfy the inequality.)
Interval \((2, 4)\) (test point \(x = 3\)): \[ (3 + 1)(3 - 2)(3 - 4) = (4)(1)(-1) = -4 < 0 \] (This interval satisfies the inequality.)
Interval \((4, \infty)\) (test point \(x = 5\)): \[ (5 + 1)(5 - 2)(5 - 4) = (6)(3)(1) = 18 > 0 \] (This interval does not satisfy the inequality.)
Consider the critical points: The expression is zero at \(x = -1\), \(x = 2\), and \(x = 4\), which means it does not satisfy the strict inequality \(< 0\) at these points.
Combine the results: The intervals where the product is negative are: \[ (-\infty, -1) \quad \text{and} \quad (2, 4) \]
Thus, the solution for the inequality \((x+1)(x-2)(x-4) < 0\) is: \[ (-\infty, -1) \cup (2, 4) \]
