We are given the compound inequality:

\[
-\frac{1}{5y} - \frac{1}{3} < \frac{2}{3}
\]

### Step 1: Isolate the term involving \( y \)
First, let's add \( \frac{1}{3} \) to both sides of the inequality to isolate the term with \( y \):

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

### Step 2: Simplify the right-hand side
Now, simplify the right-hand side by adding the fractions:

\[
\frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1
\]

So, the inequality becomes:

\[
-\frac{1}{5y} < 1
\]

### Step 3: Multiply both sides by \(-1\) and reverse the inequality
Since we're multiplying by a negative number, the inequality sign will reverse:

\[
\frac{1}{5y} > -1
\]

### Step 4: Multiply both sides by \( 5y \)
Now, multiply both sides by \( 5y \) to eliminate the fraction. But before doing this, note that we need to consider two cases: \( y > 0 \) and \( y < 0 \).

#### Case 1: \( y > 0 \)
When \( y > 0 \), multiplying by \( 5y \) maintains the inequality sign:

\[
1 > -5y
\]

Dividing both sides by \(-5\) reverses the inequality again:

\[
y > -\frac{1}{5}
\]

#### Case 2: \( y < 0 \)
When \( y < 0 \), multiplying by \( 5y \) reverses the inequality:

\[
1 < -5y
\]

Dividing by \(-5\):

\[
y < -\frac{1}{5}
\]

### Conclusion:
Thus, depending on whether \( y \) is positive or negative, the solution is:

\[
y > -\frac{1}{5} \quad \text{(if \( y > 0 \))}, \quad \text{or} \quad y < -\frac{1}{5} \quad \text{(if \( y < 0 \))}.
\]

Would you like further clarification on any of these steps?

Here are 5 follow-up questions related to inequalities and solving equations:

1. What are the rules for reversing inequality signs?
2. How do you solve inequalities involving fractions and variables?
3. Can inequalities have more than one solution?
4. What happens when you multiply both sides of an inequality by a negative number?
5. How do you solve compound inequalities with more than two parts?

**Tip**: When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Learn how to solve the compound inequality -1/5y - 1/3 < 2/3 with a step-by-step breakdown. This guide covers essential algebraic concepts, including handling inequalities, adding fractions, and multiplying by variables. Suitable for students in Grades 9-11.

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