To solve the inequality:

\[
\frac{-2}{3}(x - 3) < \frac{1}{3}(x + 8)
\]

1. **Eliminate the fractions**: Multiply both sides of the inequality by 3 to eliminate the denominators:

\[
-2(x - 3) < 1(x + 8)
\]

This simplifies to:

\[
-2x + 6 < x + 8
\]

2. **Move all terms involving \(x\) to one side**: Add \(2x\) to both sides:

\[
6 < 3x + 8
\]

3. **Isolate the \(x\) term**: Subtract 8 from both sides:

\[
-2 < 3x
\]

4. **Solve for \(x\)**: Divide both sides by 3:

\[
x > \frac{-2}{3}
\]

So the solution is:

\[
x > \frac{-2}{3}
\]

### Final Answer:
\[
x > -\frac{2}{3}
\]

Would you like further clarification or more details on any steps?

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Here are 5 related questions to further explore:

1. How would the solution change if the inequality had been \( \geq \) instead of \( < \)?
2. What if one side of the inequality involved a quadratic expression?
3. How do you solve a system of inequalities involving both \( < \) and \( > \)?
4. How do you graph the solution to this inequality on a number line?
5. What would happen if we changed the signs of the coefficients in the original equation?

**Tip:** Always check your solution by substituting values into the original inequality to ensure accuracy!

Learn how to solve the linear inequality (-2/3)(x-3) < (1/3)(x+8) using step-by-step methods. This problem involves key algebraic concepts, including eliminating fractions, isolating variables, and applying the properties of inequalities. Ideal for students in Grades 8-10.

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