https://file.aiquickdraw.com/mathbot/file/1727466051388w72g5cyn.jpg

The inequality you have uploaded is:

\[
x^2 - 5x + 6 \leq 0
\]

To solve this quadratic inequality, let's follow these steps:

### Step 1: Solve the corresponding quadratic equation
First, solve the equation:

\[
x^2 - 5x + 6 = 0
\]

This can be factored as:

\[
(x - 2)(x - 3) = 0
\]

So, the solutions to the equation are:

\[
x = 2 \quad \text{or} \quad x = 3
\]

### Step 2: Test the intervals
Now that we know the roots, \(x = 2\) and \(x = 3\), the real line is divided into three intervals:
1. \( (-\infty, 2) \)
2. \( (2, 3) \)
3. \( (3, \infty) \)

We'll test each of these intervals to see where the inequality \(x^2 - 5x + 6 \leq 0\) holds.

### Step 3: Determine the sign of the quadratic in each interval
Let's test the sign of the quadratic expression \(x^2 - 5x + 6\) in each of the intervals:

- For \(x = 1\) (in the interval \( (-\infty, 2) \)):
  \[
  1^2 - 5(1) + 6 = 1 - 5 + 6 = 2 \quad (\text{positive})
  \]

- For \(x = 2.5\) (in the interval \( (2, 3) \)):
  \[
  2.5^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25 \quad (\text{negative})
  \]

- For \(x = 4\) (in the interval \( (3, \infty) \)):
  \[
  4^2 - 5(4) + 6 = 16 - 20 + 6 = 2 \quad (\text{positive})
  \]

### Step 4: Conclusion
The quadratic expression is negative (or zero) in the interval \( [2, 3] \). Since the inequality is \( \leq 0 \), we include the endpoints.

Thus, the solution to the inequality is:

\[
2 \leq x \leq 3
\]

Would you like further details or clarifications? Here are some related questions to explore:

1. How do we graph a quadratic inequality on a number line?
2. What if the inequality was \(x^2 - 5x + 6 > 0\) instead?
3. Can all quadratic inequalities be solved by factoring?
4. How do we solve quadratic inequalities that cannot be factored?
5. What is the significance of testing intervals when solving inequalities?

**Tip:** Always check the critical points (solutions to the equation) and test the intervals in between to determine where the inequality holds.

Learn how to solve the quadratic inequality x^2 - 5x + 6 ≤ 0 using factoring and sign testing. Detailed step-by-step solution for students in Grades 8-10.

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