Let's solve the inequality $x^2 - 5x + 6 < 2$.

Step 1: Rearrange the inequality

First, we need to move all terms to one side of the inequality to simplify it:

$x^2 - 5x + 6 - 2 < 0$

This simplifies to:

$x^2 - 5x + 4 < 0$

Step 2: Solve the corresponding equation

Next, solve the quadratic equation $x^2 - 5x + 4 = 0$ to find the critical points. We can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 1$, $b = -5$, and $c = 4$.

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(4)}}{2(1)}$

$x = \frac{5 \pm \sqrt{25 - 16}}{2}$

$x = \frac{5 \pm \sqrt{9}}{2}$

$x = \frac{5 \pm 3}{2}$

So, the solutions are:

$x = \frac{5 + 3}{2} = 4 \quad \text{and} \quad x = \frac{5 - 3}{2} = 1$

Step 3: Determine the intervals

The roots divide the number line into three intervals: $(-\infty, 1)$, $(1, 4)$, and $(4, \infty)$.

Step 4: Test the intervals

We need to determine where the quadratic expression $x^2 - 5x + 4$ is negative.

1. For the interval $(-\infty, 1)$: Choose $x = 0$, $0^2 - 5(0) + 4 = 4 > 0$ So, it’s positive in this interval.

2. For the interval $(1, 4)$: Choose $x = 2$, $2^2 - 5(2) + 4 = 4 - 10 + 4 = -2 < 0$ So, it’s negative in this interval.

3. For the interval $(4, \infty)$: Choose $x = 5$, $5^2 - 5(5) + 4 = 25 - 25 + 4 = 4 > 0$ So, it’s positive in this interval.

Step 5: Conclusion

The solution to the inequality $x^2 - 5x + 4 < 0$ is:

$\boxed{1 < x < 4}$

Do you have any questions or would you like more details?

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Here are five related questions you might find interesting:

1. How do you solve quadratic inequalities using the factorization method?
2. What is the significance of the discriminant in quadratic equations?
3. How do you graph quadratic inequalities on a number line?
4. How do you solve absolute value inequalities?
5. What are the differences between solving quadratic equations and quadratic inequalities?

Tip: Always test the intervals determined by the roots to see where the inequality holds true.