Let's solve the inequality:

\[
\frac{1}{3}x - \frac{1}{4}(2x - 5) < \frac{1}{5}
\]

### Step 1: Distribute \(\frac{1}{4}\) over \((2x - 5)\)

\[
\frac{1}{4}(2x - 5) = \frac{1}{4} \times 2x - \frac{1}{4} \times 5 = \frac{2x}{4} - \frac{5}{4} = \frac{x}{2} - \frac{5}{4}
\]

Now the inequality becomes:

\[
\frac{1}{3}x - \left(\frac{x}{2} - \frac{5}{4}\right) < \frac{1}{5}
\]

### Step 2: Simplify the left-hand side

\[
\frac{1}{3}x - \frac{x}{2} + \frac{5}{4} < \frac{1}{5}
\]

### Step 3: Find a common denominator for the \(x\)-terms

The least common denominator of 3 and 2 is 6. So we rewrite the terms with a denominator of 6:

\[
\frac{1}{3}x = \frac{2}{6}x, \quad \frac{x}{2} = \frac{3}{6}x
\]

Now, substitute into the inequality:

\[
\frac{2}{6}x - \frac{3}{6}x + \frac{5}{4} < \frac{1}{5}
\]

### Step 4: Combine the \(x\)-terms

\[
\frac{2}{6}x - \frac{3}{6}x = -\frac{1}{6}x
\]

So the inequality becomes:

\[
-\frac{1}{6}x + \frac{5}{4} < \frac{1}{5}
\]

### Step 5: Eliminate the fractions by multiplying the entire inequality by 60 (the least common multiple of 6, 4, and 5)

\[
60 \times \left(-\frac{1}{6}x\right) + 60 \times \frac{5}{4} < 60 \times \frac{1}{5}
\]

Simplifying:

\[
-10x + 75 < 12
\]

### Step 6: Solve for \(x\)

First, subtract 75 from both sides:

\[
-10x < 12 - 75
\]
\[
-10x < -63
\]

Now, divide both sides by \(-10\), remembering to reverse the inequality sign:

\[
x > \frac{63}{10}
\]

### Final Solution:
\[
x > 6.3
\]

Would you like more details on any of the steps?

### Related Questions:
1. How do you solve inequalities involving fractions?
2. What happens when you multiply an inequality by a negative number?
3. How can you check the solution to an inequality?
4. How do you find a common denominator for rational expressions?
5. What is the difference between solving equations and inequalities?

### Tip:
When multiplying or dividing by a negative number in an inequality, always remember to reverse the inequality sign!

Learn how to solve the inequality 1/3x - 1/4(2x - 5) < 1/5 using fraction operations and inequality rules. This step-by-step solution demonstrates distributing fractions and solving linear inequalities. Suitable for students in Grades 9-10.

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