1. **State the problem:** Solve the inequality $x^2 - 5x + 6 < 0$. 2. **Identify the type of inequality:** This is a quadratic inequality of the form $ax^2 + bx + c < 0$. 3. **Find the roots of the corresponding quadratic equation:** Solve $x^2 - 5x + 6 = 0$. 4. **Factor the quadratic:** $x^2 - 5x + 6 = (x - 2)(x - 3)$. 5. **Set each factor equal to zero to find roots:** $$x - 2 = 0 \Rightarrow x = 2$$ $$x - 3 = 0 \Rightarrow x = 3$$ 6. **Determine intervals to test:** The roots divide the number line into three intervals: $(-\infty, 2)$, $(2, 3)$, and $(3, \infty)$. 7. **Test each interval:** - For $x = 1$ in $(-\infty, 2)$: $(1 - 2)(1 - 3) = (-1)(-2) = 2 > 0$ (does not satisfy inequality). - For $x = 2.5$ in $(2, 3)$: $(2.5 - 2)(2.5 - 3) = (0.5)(-0.5) = -0.25 < 0$ (satisfies inequality). - For $x = 4$ in $(3, \infty)$: $(4 - 2)(4 - 3) = (2)(1) = 2 > 0$ (does not satisfy inequality). 8. **Write the solution:** The inequality $x^2 - 5x + 6 < 0$ holds for $x$ in the interval $(2, 3)$. **Final answer:** $$\boxed{(2, 3)}$$