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.

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.