https://file.aiquickdraw.com/mathbot/file/1725463534510ohukakfd.jpg

The image shows a system of inequalities:

\[
\begin{cases}
x^2 - 3x \geq 0 \\
x - 4 < 0
\end{cases}
\]

Let's solve this system step by step.

### 1. Solve the first inequality: \(x^2 - 3x \geq 0\)

Factor the quadratic expression:

\[
x(x - 3) \geq 0
\]

To determine where the product is non-negative, analyze the sign changes:
- \(x(x - 3) = 0\) when \(x = 0\) or \(x = 3\).
- The sign of the expression depends on the intervals determined by these roots:
  - For \(x < 0\), \(x(x - 3)\) is positive.
  - For \(0 < x < 3\), \(x(x - 3)\) is negative.
  - For \(x > 3\), \(x(x - 3)\) is positive.

Thus, \(x^2 - 3x \geq 0\) holds for \(x \leq 0\) or \(x \geq 3\).

### 2. Solve the second inequality: \(x - 4 < 0\)

This simplifies to:

\[
x < 4
\]

### 3. Combine the results

Now, we need to find the intersection of the solution sets for both inequalities:

- From \(x^2 - 3x \geq 0\), we have \(x \leq 0\) or \(x \geq 3\).
- From \(x - 4 < 0\), we have \(x < 4\).

### Intersection of the solution sets:

- \(x \leq 0\) is valid within the range \(x < 4\), so \(x \leq 0\).
- \(x \geq 3\) is valid within the range \(x < 4\), so \(3 \leq x < 4\).

**Final Solution:**
\[
x \in (-\infty, 0] \cup [3, 4)
\]

This is the solution set for the system of inequalities.

Would you like more details or have any questions about this solution? Here are five related questions that could deepen your understanding:

1. How would you solve the inequality \(x^2 - 3x > 0\) without the equality part?
2. What is the significance of the roots of the quadratic equation in solving inequalities?
3. How do we approach systems of inequalities with more than two conditions?
4. What happens if the inequalities in the system are strict (i.e., using \(>\) instead of \(\geq\))?
5. Can you graph the solution set on a number line?

**Tip:** When dealing with inequalities involving quadratic expressions, always consider factoring as a first step to determine sign changes across different intervals.

Solve the system of inequalities: x^2 - 3x ≥ 0 and x - 4 < 0.

Learn how to solve a system of inequalities involving a quadratic and linear inequality. This problem covers algebraic concepts such as factoring quadratic expressions and finding intersections of solution sets. Suitable for students in Grades 9-10.

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