The inequality you provided is:

$\left| \frac{x - 1}{x} - 3 \right| > 6$

This is an absolute value inequality, and we'll solve it step by step.

Step 1: Rewrite the inequality without the absolute value

The absolute value inequality $\left| A \right| > B$ can be rewritten as two inequalities:

$A > B \quad \text{or} \quad A < -B$

In this case:

$\frac{x - 1}{x} - 3 > 6 \quad \text{or} \quad \frac{x - 1}{x} - 3 < -6$

Step 2: Solve each inequality

Inequality 1: $\frac{x - 1}{x} - 3 > 6$

1. Add 3 to both sides: $\frac{x - 1}{x} > 9$

2. Multiply both sides by $x$ (assuming $x \neq 0$): $x - 1 > 9x$

3. Rearrange the terms: $-1 > 8x$

4. Solve for $x$: $x < -\frac{1}{8}$

Inequality 2: $\frac{x - 1}{x} - 3 < -6$

1. Add 3 to both sides: $\frac{x - 1}{x} < -3$

2. Multiply both sides by $x$ (assuming $x \neq 0$): $x - 1 < -3x$

3. Rearrange the terms: $4x < 1$

4. Solve for $x$: $x < \frac{1}{4}$

Step 3: Combine the two results

We now combine the two cases:

1. From the first inequality: $x < -\frac{1}{8}$
2. From the second inequality: $x < \frac{1}{4}$

But the second inequality also implies that the solution must be less than $\frac{1}{4}$, so the overall solution is:

$x < -\frac{1}{8} \quad \text{or} \quad x > \frac{1}{4}$

Final solution:

The solution to the inequality is:

$x < -\frac{1}{8} \quad \text{or} \quad x > \frac{1}{4}$

Would you like more details or have any further questions?

Here are some related questions:

1. How do you solve absolute value inequalities in general?
2. What are the conditions when multiplying inequalities by variables?
3. Can you graph the solution on a number line?
4. How does the inequality change if we have $\leq$ or $\geq$ instead of $\gt$?
5. What are the steps to solve compound inequalities?

Tip: Always be cautious when multiplying inequalities by variables, especially when you don't know their sign, as it can reverse the inequality!