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### 1. \( x^2 - 10x + 16 < 0 \)

We can solve this quadratic inequality by factoring or using the quadratic formula. Let's solve it.

The equation is \( x^2 - 10x + 16 = 0 \). First, we find the discriminant:

\[
\Delta = (-10)^2 - 4(1)(16) = 100 - 64 = 36
\]

So, the roots are:

\[
x = \frac{-(-10) \pm \sqrt{36}}{2(1)} = \frac{10 \pm 6}{2}
\]

\[
x = \frac{10 + 6}{2} = 8 \quad \text{or} \quad x = \frac{10 - 6}{2} = 2
\]

Thus, the quadratic factors as:

\[
(x - 2)(x - 8) < 0
\]

To solve this inequality, we check the sign of the product \( (x - 2)(x - 8) \) in the intervals divided by the roots 2 and 8:

- For \( x < 2 \), both factors are negative, so the product is positive.
- For \( 2 < x < 8 \), one factor is positive and the other is negative, so the product is negative.
- For \( x > 8 \), both factors are positive, so the product is positive.

Thus, the solution is \( 2 < x < 8 \).

**Answer in interval notation**: \( (2, 8) \).

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Would you like me to continue with the next problems or explain anything in more detail? 

Here are five related problems for practice:

1. Solve \( x^2 - 12x + 32 < 0 \).
2. Solve \( x^2 + 6x + 9 \leq 0 \).
3. Find the solution to \( x^2 - 3x - 4 > 0 \).
4. Solve \( x^2 - 2x - 8 \geq 0 \) and graph the solution.
5. Factor and solve \( x^2 + 8x + 16 = 0 \).

**Tip**: For quadratic inequalities, always check the sign of the expression between the roots to determine where the inequality holds.

Solve, graph on a number line, and write your answer in interval notation: x^2 - 10x + 16 < 0.

Learn how to solve and graph the quadratic inequality x^2 - 10x + 16 < 0 using factoring and interval analysis. This example walks through solving quadratic inequalities by identifying roots and determining the intervals where the inequality holds. Suitable for high school students studying algebra.

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