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We are tasked with finding all the integer solutions for the system of inequalities:

\[
\begin{cases} 
3 - 18x < 0 \\
0.2 - 0.1x > 0 
\end{cases}
\]

### Step 1: Solve each inequality separately.

**First inequality: \( 3 - 18x < 0 \)**

\[
3 - 18x < 0
\]
Subtract 3 from both sides:
\[
-18x < -3
\]
Now divide both sides by -18. Remember that dividing by a negative number reverses the inequality:
\[
x > \frac{-3}{-18} = \frac{1}{6}
\]
So, \( x > \frac{1}{6} \).

**Second inequality: \( 0.2 - 0.1x > 0 \)**

\[
0.2 - 0.1x > 0
\]
Subtract 0.2 from both sides:
\[
-0.1x > -0.2
\]
Now divide by -0.1, reversing the inequality:
\[
x < \frac{-0.2}{-0.1} = 2
\]
So, \( x < 2 \).

### Step 2: Combine the solutions.

The solutions are \( x > \frac{1}{6} \) and \( x < 2 \). Combining these, we get:
\[
\frac{1}{6} < x < 2
\]

### Step 3: Find integer solutions.

The integers between \( \frac{1}{6} \) and 2 are \( x = 1 \).

### Final Answer:
The only integer solution is \( x = 1 \).

---

Would you like more details on any step? Here are 5 related questions for further practice:

1. What happens if we reverse both inequalities?
2. How do you solve systems of inequalities with more than two inequalities?
3. Can you graph the solution to this system of inequalities?
4. What would the solution be if one of the inequalities had a "greater than or equal to" symbol?
5. How would you solve these inequalities if they involved absolute values?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Find all the integer solutions of the system of inequalities: { 3 - 18x < 0, 0.2 - 0.1x > 0 }

Learn how to solve the system of linear inequalities 3 - 18x < 0 and 0.2 - 0.1x > 0, with step-by-step solutions. The problem uses algebraic manipulation of inequalities and covers integer solutions suitable for students in Grades 9-11.

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