Solve the inequality (x - 2)(x + 1)(x - 4) < 0.

Step-by-step solution

1. 1

   Find the critical points. The product is zero exactly where a factor vanishes:

   $x=-1,\qquad x=2,\qquad x=4.$

   Because the inequality is strict, none of these three values belongs to the solution set — they are boundaries only. Ordering them from smallest to largest is essential before drawing the chart.

2. 2

   Split the number line into four intervals. The three roots create

   $(-\infty,-1),\quad(-1,2),\quad(2,4),\quad(4,\infty).$

   The sign of a product of linear factors is constant on each interval, so a single test point per interval settles it.

3. 3

   Test one point in each interval. Count the negative factors:

   $x=-2:\ (-4)(-1)(-6)<0;\qquad x=0:\ (-2)(1)(-4)>0;$

   $x=3:\ (1)(4)(-1)<0;\qquad x=5:\ (3)(6)(1)>0.$

   An odd number of negative factors makes the product negative, an even number makes it positive — which is why the signs alternate as you cross each root.

4. 4

   Use the multiplicity shortcut as a check. Every root here is simple (multiplicity one), so the sign flips at each one. Starting from the far right, where all three factors are positive, the pattern going leftwards must be $+,-,+,-$ — exactly matching the test points ✓. Had any factor been squared, the sign would not have flipped there.

5. 5

   Collect the intervals where the product is negative. Reading the chart:

   $x\in(-\infty,-1)\cup(2,4).$

   All endpoints are open. If the inequality had been $\le0$, the answer would instead be $(-\infty,-1]\cup[2,4]$, with the three roots included.