The inequality given is:

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

To solve this absolute value inequality, we need to break it into two cases based on the definition of absolute value.

Case 1:

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

Case 2:

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

Solving Case 1:

$\frac{1}{x} - 3 > 6$ Add 3 to both sides: $\frac{1}{x} > 9$ Take the reciprocal of both sides (note that taking the reciprocal reverses the inequality): $x < \frac{1}{9}$ So, for Case 1, $x < \frac{1}{9}$.

Solving Case 2:

$\frac{1}{x} - 3 < -6$ Add 3 to both sides: $\frac{1}{x} < -3$ Take the reciprocal of both sides (again, taking the reciprocal reverses the inequality): $x > -\frac{1}{3}$ So, for Case 2, $x > -\frac{1}{3}$.

Final Solution:

The solution is the union of the two cases: $x < \frac{1}{9} \quad \text{or} \quad x > -\frac{1}{3}$

However, because $x$ cannot be zero (since it would make the original expression undefined), the solution is:

$x \in \left(-\infty, -\frac{1}{3}\right) \cup \left(0, \frac{1}{9}\right)$

Would you like more details or have any questions?

Related Questions:

1. How do you solve absolute value equations?
2. What happens when taking the reciprocal of an inequality?
3. What are some common mistakes in solving absolute value inequalities?
4. How do you graph the solution set on a number line?
5. Can this method be applied to more complex absolute value inequalities?

Tip:

When solving inequalities involving absolute values, always remember to consider both the positive and negative scenarios.