The inequality provided is:

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

To solve this inequality, let's break it down:

Step 1: Remove the absolute value by considering two cases

The absolute value inequality $\left| A \right| > B$ can be split into two cases:

1. $A > B$
2. $A < -B$

In this case, $A = \frac{1}{x} - 3$ and $B = 6$.

Case 1:

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

Add 3 to both sides:

$\frac{1}{x} > 9$

Now, take the reciprocal of both sides (remember that taking the reciprocal reverses the inequality sign):

$x < \frac{1}{9}$

Case 2:

$\frac{1}{x} - 3 < -6$

Add 3 to both sides:

$\frac{1}{x} < -3$

Take the reciprocal of both sides:

$x > -\frac{1}{3}$

Step 2: Combine the solutions

From the two cases, we get two intervals:

• $x < \frac{1}{9}$
• $x > -\frac{1}{3}$

Step 3: Find the union of these intervals

The solution to the inequality is:

$x \in \left(-\frac{1}{3}, \frac{1}{9}\right)$

Final Answer:

The solution is $x \in \left(-\frac{1}{3}, \frac{1}{9}\right)$, meaning $x$ lies between $-\frac{1}{3}$ and $\frac{1}{9}$, but does not include these endpoints.

Would you like further details on any part of this process or have any additional questions?

Here are some related questions you might find helpful:

1. How do you solve absolute value inequalities in general?
2. What are the key steps in solving rational inequalities?
3. How do you deal with inequalities involving reciprocals?
4. Can you explain how to graph the solution on a number line?
5. How do you determine the critical points when solving inequalities?

Tip: When solving absolute value inequalities, always remember to consider both the positive and negative scenarios to get the complete solution.