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To solve the inequality \( 6x^2 - 3x < 0 \), follow these steps:

### Step 1: Factor the inequality
Factor the quadratic expression on the left-hand side.

\[
6x^2 - 3x = 3x(2x - 1)
\]

So, the inequality becomes:

\[
3x(2x - 1) < 0
\]

### Step 2: Find the critical points
To solve \( 3x(2x - 1) = 0 \), set each factor equal to zero:

1. \( 3x = 0 \)  ⟶  \( x = 0 \)
2. \( 2x - 1 = 0 \) ⟶ \( x = \frac{1}{2} \)

These values, \( x = 0 \) and \( x = \frac{1}{2} \), are the critical points where the expression changes sign.

### Step 3: Determine the sign in each interval
The critical points divide the number line into three intervals:

1. \( x < 0 \)
2. \( 0 < x < \frac{1}{2} \)
3. \( x > \frac{1}{2} \)

Now, choose a test point from each interval and determine whether the product \( 3x(2x - 1) \) is positive or negative in that interval.

- For \( x < 0 \) (e.g., \( x = -1 \)):  
  \( 3(-1)(2(-1) - 1) = 3(-1)(-3) = 9 \), which is **positive**.

- For \( 0 < x < \frac{1}{2} \) (e.g., \( x = \frac{1}{4} \)):  
  \( 3\left(\frac{1}{4}\right)\left(2\left(\frac{1}{4}\right) - 1\right) = 3\left(\frac{1}{4}\right)\left(\frac{1}{2} - 1\right) = 3\left(\frac{1}{4}\right)\left(-\frac{1}{2}\right) = -\frac{3}{8} \), which is **negative**.

- For \( x > \frac{1}{2} \) (e.g., \( x = 1 \)):  
  \( 3(1)(2(1) - 1) = 3(1)(1) = 3 \), which is **positive**.

### Step 4: Write the solution
We want the product \( 3x(2x - 1) \) to be **negative**, so the inequality is satisfied in the interval where the expression is negative, which is \( 0 < x < \frac{1}{2} \).

### Final answer:
\[
0 < x < \frac{1}{2}
\]

Would you like more details or have any questions about this solution?

### Related questions:
1. How do you solve quadratic inequalities by factoring?
2. What does it mean when an inequality involves a quadratic expression?
33. How do you graph the solution to a quadratic inequality on a number line?
4. What are critical points, and how do they affect the solution to an inequality?
5. How can you verify the solution to an inequality using test points?

### Tip:
When solving inequalities involving polynomials, always factor the expression first and then test intervals between the critical points to determine where the inequality holds.

Learn how to solve the quadratic inequality 6x^2 - 3x < 0 with step-by-step factoring. This solution covers key algebra concepts such as finding critical points, testing intervals, and applying the zero product property. Ideal for students in Grades 9-11.

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