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Let's solve these inequalities step-by-step.

### Problem 1
The inequalities given are:
1. \( x^2 - 8x + 12 < 0 \)
2. \( x^2 - 6x - 7 \geq 0 \)

#### Part (a): \( x^2 - 8x + 12 < 0 \)
1. **Factorize the quadratic expression**: 
   \[
   x^2 - 8x + 12 = (x - 6)(x - 2)
   \]
   
2. **Set up the inequality**:
   \[
   (x - 6)(x - 2) < 0
   \]

3. **Determine intervals**:
   The roots of the equation are \( x = 6 \) and \( x = 2 \). These points divide the number line into three intervals:
   - \( x < 2 \)
   - \( 2 < x < 6 \)
   - \( x > 6 \)

4. **Test each interval**:
   - For \( x < 2 \), both factors \( (x - 6) \) and \( (x - 2) \) are negative, so their product is positive.
   - For \( 2 < x < 6 \), \( (x - 6) \) is negative and \( (x - 2) \) is positive, so their product is negative.
   - For \( x > 6 \), both factors are positive, so their product is positive.

5. **Conclusion**:
   The solution for \( (x - 6)(x - 2) < 0 \) is:
   \[
   2 < x < 6
   \]

#### Part (b): \( x^2 - 6x - 7 \geq 0 \)
1. **Factorize the quadratic expression**:
   \[
   x^2 - 6x - 7 = (x - 7)(x + 1)
   \]

2. **Set up the inequality**:
   \[
   (x - 7)(x + 1) \geq 0
   \]

3. **Determine intervals**:
   The roots of the equation are \( x = 7 \) and \( x = -1 \). These points divide the number line into three intervals:
   - \( x < -1 \)
   - \( -1 < x < 7 \)
   - \( x > 7 \)

4. **Test each interval**:
   - For \( x < -1 \), both factors \( (x - 7) \) and \( (x + 1) \) are negative, so their product is positive.
   - For \( -1 < x < 7 \), \( (x - 7) \) is negative and \( (x + 1) \) is positive, so their product is negative.
   - For \( x > 7 \), both factors are positive, so their product is positive.

5. **Conclusion**:
   The solution for \( (x - 7)(x + 1) \geq 0 \) is:
   \[
   x \leq -1 \text{ or } x \geq 7
   \]

### Final Answer:
- **Part (a)**: \( 2 < x < 6 \)
- **Part (b)**: \( x \leq -1 \) or \( x \geq 7 \)

Would you like a detailed explanation of each step? Here are some follow-up questions to explore further:

1. How do we factorize a quadratic expression?
2. Why do we test intervals around the roots?
3. Can you solve a similar inequality with different coefficients?
4. What is the significance of finding intervals for the inequality?
5. How would the solution change if the inequality sign were reversed?

**Tip**: When solving quadratic inequalities, always factorize first and use interval testing based on the roots.

Selesaikan pertidaksamaan berikut ini:
1. x^2 - 8x + 12 < 0
2. x^2 - 6x - 7 ≥ 0

Learn how to solve the quadratic inequalities x^2 - 8x + 12 < 0 and x^2 - 6x - 7 ≥ 0 with a detailed solution. This problem involves factoring, interval testing, and quadratic inequality concepts, suitable for high school students in Grades 10-12.

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