1. **State the problem:** Solve the inequality $$(x - 2)(x - 4)(x - 1) < 0$$ 2. **Identify the roots:** The expression equals zero at $x=1$, $x=2$, and $x=4$. These points divide the number line into four intervals: $(-\infty,1)$, $(1,2)$, $(2,4)$, and $(4,\infty)$. 3. **Determine the sign of each factor in each interval:** - For $x < 1$: $(x-1)<0$, $(x-2)<0$, $(x-4)<0$ so product is $(-)(-)(-) = -$ (negative). - For $1 < x < 2$: $(x-1)>0$, $(x-2)<0$, $(x-4)<0$ so product is $(+)(-)(-) = +$ (positive). - For $2 < x < 4$: $(x-1)>0$, $(x-2)>0$, $(x-4)<0$ so product is $(+)(+)(-) = -$ (negative). - For $x > 4$: $(x-1)>0$, $(x-2)>0$, $(x-4)>0$ so product is $(+)(+)(+) = +$ (positive). 4. **Find where the product is less than zero:** This happens where the product is negative, i.e., on intervals $(-\infty,1)$ and $(2,4)$. 5. **Check if roots are included:** Since the inequality is strict ($<$), roots where the product equals zero are not included. **Final answer:** $$x \in (-\infty,1) \cup (2,4)$$