1. **State the problem:** Solve the inequality $ (x - 4)(x - 2) > 0 $. 2. **Recall the rule:** A product of two factors is positive if both factors are positive or both are negative. 3. **Find critical points:** Set each factor equal to zero: $$ x - 4 = 0 \Rightarrow x = 4 $$ $$ x - 2 = 0 \Rightarrow x = 2 $$ These points divide the number line into three intervals: $(-\infty, 2)$, $(2, 4)$, and $(4, \infty)$. 4. **Test each interval:** - For $x < 2$, pick $x=0$: $$ (0-4)(0-2) = (-4)(-2) = 8 > 0 $$ - For $2 < x < 4$, pick $x=3$: $$ (3-4)(3-2) = (-1)(1) = -1 < 0 $$ - For $x > 4$, pick $x=5$: $$ (5-4)(5-2) = (1)(3) = 3 > 0 $$ 5. **Write solution:** The inequality holds where the product is positive, so $$ x \in (-\infty, 2) \cup (4, \infty) $$ 6. **Final answer:** $$ \boxed{x < 2 \text{ or } x > 4} $$