https://file.aiquickdraw.com/mathbot/file/17291743823706yo3efxa.jpg

The image shows the mathematical inequality:

\[
\frac{3}{2x} \ > \ \frac{5}{6x^2}
\]

We need to solve for \( x \). Here's how we can approach it:

### Step 1: Eliminate the denominators by multiplying both sides by the least common denominator (LCD).
The denominators are \( 2x \) and \( 6x^2 \). The LCD is \( 6x^2 \).

Multiply both sides of the inequality by \( 6x^2 \):

\[
6x^2 \times \frac{3}{2x} > 6x^2 \times \frac{5}{6x^2}
\]

### Step 2: Simplify both sides.
- On the left side: \( 6x^2 \times \frac{3}{2x} = 3x \times 3 = 9x \)
- On the right side: \( 6x^2 \times \frac{5}{6x^2} = 5 \)

So the inequality becomes:

\[
9x > 5
\]

### Step 3: Solve for \( x \).
To isolate \( x \), divide both sides by 9:

\[
x > \frac{5}{9}
\]

### Final answer:
The solution to the inequality is \( x > \frac{5}{9} \).

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. What happens to the inequality if \( x \) were negative?
2. How would you solve this if the inequality were \( \frac{3}{2x} < \frac{5}{6x^2} \)?
3. Can you find the value of \( x \) where both sides of the inequality are equal?
4. How would you represent this solution on a number line?
5. What would the inequality look like if we had different constants in the numerators?

**Tip:** Always be cautious when multiplying or dividing inequalities by negative values, as this reverses the inequality sign.

Learn how to solve the rational inequality \(\frac{3}{2x} > \frac{5}{6x^2}\) with step-by-step calculations. This problem covers algebraic concepts like inequalities, rational expressions, and simplification methods. Ideal for students in Grades 8-10.

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