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To solve the inequality:

\[
x - 2 < 3(x - 5)
\]

we’ll go through the following steps:

### Step 1: Distribute the 3 on the right side
\[
x - 2 < 3x - 15
\]

### Step 2: Move all terms involving \( x \) to one side
Subtract \( x \) from both sides:
\[
-2 < 2x - 15
\]

### Step 3: Isolate the term with \( x \)
Add 15 to both sides:
\[
13 < 2x
\]

### Step 4: Solve for \( x \) by dividing both sides by 2
\[
\frac{13}{2} < x
\]
or equivalently,
\[
x > \frac{13}{2}
\]

### Final Answer:
\[
x > \frac{13}{2} \quad \text{or} \quad x > 6.5
\]

Would you like further details on any of the steps?

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Here are five related questions to explore further:

1. How would the solution change if the inequality was \( x - 2 \leq 3(x - 5) \)?
2. What happens to the inequality direction when multiplying or dividing by a negative number?
3. Can you solve similar inequalities where the terms include square roots or fractions?
4. How would the inequality be solved if it were a quadratic inequality instead?
5. What graphical representation corresponds to this inequality on a number line?

**Tip:** When solving inequalities, always remember to flip the inequality sign if you multiply or divide both sides by a negative number.

Learn how to solve the linear inequality x - 2 < 3(x - 5) with a detailed step-by-step solution. This problem demonstrates key concepts in inequalities and linear equations, suitable for students in Grades 8-10.

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