Let's solve the inequality step by step:

Given: $x - 7 > 3(3x - 9)$

Step 1: Expand the right-hand side

Distribute the $3$ across the terms inside the parentheses:

$x - 7 > 3 \cdot 3x - 3 \cdot 9$

This simplifies to:

$x - 7 > 9x - 27$

Step 2: Move all terms involving $x$ to one side

To get all the $x$-terms on one side and constants on the other, subtract $x$ from both sides:

$-7 > 8x - 27$

Step 3: Move constants to the other side

Now, add $27$ to both sides:

$20 > 8x$

Step 4: Solve for $x$

To isolate $x$, divide both sides of the inequality by 8:

$x < \frac{20}{8}$

Simplify the fraction:

$x < \frac{5}{2}$

Thus, the solution is: $x < 2.5$

Conclusion:

The solution to the inequality is $x < 2.5$.

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Would you like more details on any of the steps?

Here are some related questions for further exploration:

1. How do you solve an inequality with fractions, like $\frac{1}{2}x - 3 > 2$?
2. What happens if the inequality contains a negative coefficient, like $-2x + 5 < 7$?
3. How do you solve a system of linear inequalities?
4. What is the difference between solving an equation and solving an inequality?
5. How do you graph an inequality on a number line?

Tip: Always reverse the inequality sign when multiplying or dividing both sides of an inequality by a negative number!