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Latihan Soal:
1. X^2 - 9x + 8 ≥ 0
2. X^2 - 35x + 300 ≤ 0
3. -2x^2 + 7x + 4 ≤ 0
4. 2x^2 + 7x - 10 ≤ X^2 + 12
5. -4x^2 - x + 2 ≥ 0

Here are the solutions for each of the inequalities you've provided:

1. **\( x^2 - 9x + 8 \geq 0 \)**
   Factorize: \( (x - 1)(x - 8) \geq 0 \)
   Critical points: \( x = 1 \) and \( x = 8 \)

   The solution is: \( x \leq 1 \) or \( x \geq 8 \).

2. **\( x^2 - 35x + 300 \leq 0 \)**
   Factorize: \( (x - 15)(x - 20) \leq 0 \)
   Critical points: \( x = 15 \) and \( x = 20 \)

   The solution is: \( 15 \leq x \leq 20 \).

3. **\( -2x^2 + 7x + 4 \leq 0 \)**
   Rewrite as: \( 2x^2 - 7x - 4 \geq 0 \)
   Factorize: \( (2x + 1)(x - 4) \geq 0 \)
   Critical points: \( x = -\frac{1}{2} \) and \( x = 4 \)

   The solution is: \( x \leq -\frac{1}{2} \) or \( x \geq 4 \).

4. **\( 2x^2 + 7x - 10 \leq x^2 + 12 \)**
   Simplify: \( x^2 + 7x - 22 \leq 0 \)
   Factorize: \( (x - 2)(x + 11) \leq 0 \)
   Critical points: \( x = 2 \) and \( x = -11 \)

   The solution is: \( -11 \leq x \leq 2 \).

5. **\( -4x^2 - x + 2 \geq 0 \)**
   Rewrite: \( 4x^2 + x - 2 \leq 0 \)
   Factorize: \( (4x - 3)(x + 2) \leq 0 \)
   Critical points: \( x = \frac{3}{4} \) and \( x = -2 \)

   The solution is: \( -2 \leq x \leq \frac{3}{4} \).

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Would you like a detailed breakdown of any of these steps? Here are some follow-up questions for further practice:

1. How do we determine the critical points in these inequalities?
2. Why do we change the sign of inequalities when multiplying or dividing by a negative?
3. What are the rules for solving quadratic inequalities?
4. How can graphing help visualize the solutions to inequalities?
5. How would you solve an inequality involving absolute values?

**Tip:** Always check the intervals created by critical points to determine the correct signs of the inequality in each region.

Learn how to solve quadratic and absolute value inequalities, including problems like X^2 - 9x + 8 ≥ 0 and -4x^2 - x + 2 ≥ 0. This guide covers key algebraic concepts, such as factorization and properties of real numbers, and provides a step-by-step solution for each problem. Suitable for students in Grades 10-12.

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