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?

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.