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To solve the rational inequality:

\[
\frac{3}{x} + \frac{2}{x} > 1
\]

Let's go step by step, following the instructions from the definition of rational inequalities.

### Step 1: Combine all terms into a single rational expression
Combine the terms on the left-hand side:

\[
\frac{3}{x} + \frac{2}{x} = \frac{3 + 2}{x} = \frac{5}{x}
\]

So, the inequality becomes:

\[
\frac{5}{x} > 1
\]

### Step 2: Determine the critical points
Now, let's find the critical points by solving the equality \(\frac{5}{x} = 1\) and determining where the expression is undefined (since this is a rational function).

1. Solve for \(x\) when \(\frac{5}{x} = 1\):
   \[
   \frac{5}{x} = 1 \implies 5 = x
   \]

2. The expression \(\frac{5}{x}\) is undefined when \(x = 0\), because division by zero is undefined.

Thus, the critical points are \(x = 5\) and \(x = 0\).

### Step 3: Construct a number line and divide into intervals
The critical points divide the number line into three intervals:

1. \(x < 0\)
2. \(0 < x < 5\)
3. \(x > 5\)

### Step 4: Test a value in each interval
We will test values from each interval in the inequality \(\frac{5}{x} > 1\).

1. **For \(x < 0\):** Let's test \(x = -1\):
   \[
   \frac{5}{-1} = -5 \quad (\text{since } -5 \not> 1)
   \]
   So, the inequality is **false** in this interval.

2. **For \(0 < x < 5\):** Let's test \(x = 1\):
   \[
   \frac{5}{1} = 5 \quad (\text{since } 5 > 1)
   \]
   So, the inequality is **true** in this interval.

3. **For \(x > 5\):** Let's test \(x = 6\):
   \[
   \frac{5}{6} \approx 0.83 \quad (\text{since } 0.83 \not> 1)
   \]
   So, the inequality is **false** in this interval.

### Step 5: Determine the intervals where the inequality holds
From the tests, the inequality holds in the interval \(0 < x < 5\).

### Final Answer:
The solution to the inequality is:

\[
0 < x < 5
\]

This means \(x\) must be in the interval \((0, 5)\), excluding 0 and 5.

---

Would you like more details on any step, or further examples? 

Here are some follow-up questions to practice on rational inequalities:

1. Solve: \(\frac{4}{x} - \frac{1}{x} \leq 2\)
2. Solve: \(\frac{1}{x} + 3 \geq \frac{4}{x+2}\)
3. Solve: \(\frac{6}{x-1} > 3\)
4. Solve: \(\frac{2x}{x-4} \leq 1\)
5. Solve: \(\frac{5x-1}{x+3} < 2\)

**Tip**: Always check the domain of the function (where it's undefined) before solving rational inequalities.

Solve the rational inequality: 3/x + 2/x > 1

Learn how to solve the rational inequality 3/x + 2/x > 1 using key algebraic techniques such as combining rational expressions and testing critical points. This step-by-step solution is ideal for students in Grades 9-11.

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