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The problem in the image asks: 

**Which of the following value of \(x\) do not satisfy the inequality \(x^2 - 3x + 2 > 0\) at all?**

The options are:
a) \(1 \leq x \leq 2\)  
b) \(-1 \leq x \leq -2\)  
c) \(x \leq 2\)  
d) \(0 \leq x \geq 2\)

### Solution:

We start by solving the quadratic inequality \(x^2 - 3x + 2 > 0\).

#### Step 1: Solve \(x^2 - 3x + 2 = 0\)

Factoring the quadratic equation:
\[
x^2 - 3x + 2 = (x - 1)(x - 2) = 0
\]

So, the roots of the equation are \(x = 1\) and \(x = 2\).

#### Step 2: Analyze the sign of \(x^2 - 3x + 2\)

Now, we need to determine where the quadratic expression is positive. We will check the intervals defined by the roots:

- For \(x < 1\), pick \(x = 0\):  
  \(x^2 - 3x + 2 = 0^2 - 3(0) + 2 = 2 > 0\)  
  So, the inequality is satisfied for \(x < 1\).

- For \(1 < x < 2\), pick \(x = 1.5\):  
  \(x^2 - 3x + 2 = 1.5^2 - 3(1.5) + 2 = 2.25 - 4.5 + 2 = -0.25 < 0\)  
  So, the inequality is not satisfied for \(1 < x < 2\).

- For \(x > 2\), pick \(x = 3\):  
  \(x^2 - 3x + 2 = 3^2 - 3(3) + 2 = 9 - 9 + 2 = 2 > 0\)  
  So, the inequality is satisfied for \(x > 2\).

#### Step 3: Combine results

The inequality \(x^2 - 3x + 2 > 0\) is satisfied for:
- \(x < 1\)
- \(x > 2\)

The inequality is **not satisfied** for \(1 \leq x \leq 2\).

#### Conclusion:
The correct answer is **a) \(1 \leq x \leq 2\)**, as this interval does not satisfy the inequality at all.

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Would you like a more detailed explanation or have any questions?

Here are 5 related questions for further practice:
1. Solve \(x^2 - 5x + 6 > 0\).
2. Solve the inequality \((x - 4)(x + 2) > 0\).
3. Find the values of \(x\) that satisfy \(x^2 - 4x + 3 < 0\).
4. For what values of \(x\) is \(x^2 + x - 6 \geq 0\)?
5. Solve the quadratic inequality \(2x^2 - 3x - 5 \leq 0\).

**Tip:** When solving quadratic inequalities, it helps to plot the parabola or analyze sign changes at the roots.

Which of the following value of x do not satisfy the inequality (x^2 - 3x + 2 > 0) at all?

Explore how to solve the quadratic inequality x^2 - 3x + 2 > 0 using factorization. This problem involves determining intervals where the inequality holds true, using the roots and analyzing sign changes in the quadratic expression. Suitable for students in Grades 8-10.

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