https://file.aiquickdraw.com/mathbot/file/1729135008108db0jcv0x.jpg

The problem you're working on is to solve the compound inequality:

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

### Steps to solve algebraically:

1. **Solve the inequality** \( 0 \leq x^2 - 5x + 6 \).

This is equivalent to solving the quadratic inequality:

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

This quadratic expression is non-negative (greater than or equal to 0) for values of \(x \leq 2\) and \(x \geq 3\). So the solution for this part is \(x \in (-\infty, 2] \cup [3, \infty)\).

2. **Solve the inequality** \( x^2 - 5x + 6 < 2 \).

To solve this inequality, subtract 2 from both sides:

\[
x^2 - 5x + 4 < 0
\]

Now, factor the quadratic:

\[
(x - 1)(x - 4) < 0
\]

This quadratic is less than 0 between the roots, so the solution is \(1 < x < 4\).

### Combine the Solutions:

From the first inequality \( 0 \leq x^2 - 5x + 6 \), we have \(x \in (-\infty, 2] \cup [3, \infty)\).

From the second inequality \(x^2 - 5x + 4 < 0\), we have \(x \in (1, 4)\).

Now, we take the intersection of the two sets:

\[
x \in (1, 2] \cup [3, 4)
\]

### Graphing the Solution:

On a number line, the solution set would include:
- An interval from \(1\) (open circle) to \(2\) (closed circle).
- Another interval from \(3\) (closed circle) to \(4\) (open circle).

You can now choose the correct graphical representation based on this solution.

Would you like more details or clarifications on any step?

### Relative Questions:
1. How do we graph the solution set on a number line?
2. What is the process for solving quadratic inequalities?
3. Why do we take the intersection of solution sets in compound inequalities?
4. What are some ways to factor quadratic equations quickly?
5. Can this inequality be solved using a different method like completing the square?

### Tip:
Always remember to check for intersections when dealing with compound inequalities, as you need to find where both conditions are satisfied simultaneously.

Solve 0 ≤ x^2 - 5x + 6 < 2 algebraically. Graph the solution.

Step-by-step solution to solve the quadratic inequality 0 ≤ x^2 - 5x + 6 < 2. Includes factoring quadratics and graphing the solution set on a number line. Suitable for Grades 9-11.

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