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The inequality provided is:

\[
\frac{1}{4}(2x - 1) - x < \frac{x}{6} - \frac{1}{3}
\]

### Steps to solve:

1. **Distribute** the \(\frac{1}{4}\) on the left-hand side:
   \[
   \frac{1}{4}(2x - 1) = \frac{1}{4} \cdot 2x - \frac{1}{4} \cdot 1 = \frac{2x}{4} - \frac{1}{4} = \frac{x}{2} - \frac{1}{4}
   \]
   Now, the inequality becomes:
   \[
   \frac{x}{2} - \frac{1}{4} - x < \frac{x}{6} - \frac{1}{3}
   \]

2. **Combine like terms** on the left-hand side:
   \[
   \frac{x}{2} - x = \frac{x}{2} - \frac{2x}{2} = -\frac{x}{2}
   \]
   Now, the inequality is:
   \[
   -\frac{x}{2} - \frac{1}{4} < \frac{x}{6} - \frac{1}{3}
   \]

3. **Eliminate the fractions** by finding the least common denominator (LCD) of all terms, which is 12. Multiply both sides of the inequality by 12 to clear the denominators:
   \[
   12\left(-\frac{x}{2} - \frac{1}{4}\right) < 12\left(\frac{x}{6} - \frac{1}{3}\right)
   \]
   This simplifies to:
   \[
   -6x - 3 < 2x - 4
   \]

4. **Move all terms involving \(x\)** to one side of the inequality and constants to the other:
   \[
   -6x - 2x < -4 + 3
   \]
   Simplifying both sides:
   \[
   -8x < -1
   \]

5. **Solve for \(x\)** by dividing both sides by \(-8\) (and remember to flip the inequality sign when dividing by a negative number):
   \[
   x > \frac{1}{8}
   \]

### Final Solution:
\[
x > \frac{1}{8}
\]

Let me know if you want more details or have any questions!

Here are 5 related questions:
1. How can inequalities be solved if there are variables on both sides?
2. What are the rules for flipping inequality signs when multiplying or dividing by a negative number?
3. How do you find the least common denominator when dealing with fractions in equations or inequalities?
4. Can you explain the steps to isolate variables in complex inequalities?
5. How does solving inequalities differ from solving equations?

**Tip:** Always remember to flip the inequality sign when multiplying or dividing both sides of an inequality by a negative number!

Solve the inequality: 1/4 (2x - 1) - x < x/6 - 1/3

Learn how to solve the inequality 1/4(2x - 1) - x < x/6 - 1/3 with a step-by-step explanation. This problem covers key algebraic concepts, including the distributive property, combining like terms, and handling inequalities involving fractions. Suitable for students in Grades 9-10.

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