### Instruction: Solve the inequality
\[\left| \frac{2x - 1}{x - 1} \right| > 2.\].### Response: I need to find all the values of x that make this inequality true.
First, I notice that x cannot be equal to 1, because that would make the denominator zero and the expression undefined.
So I can exclude x = 1 from the possible solutions and consider the cases where x is either less than 1 or greater than 1.
If x is less than 1, then the expression inside the absolute value is negative, so I can rewrite the inequality as
\[\frac{1 - 2x}{x - 1} > 2.\]
To solve this, I can multiply both sides by x - 1, which is also negative, and reverse the inequality sign.
I get
\[1 - 2x < 2x - 2.\]
Simplifying, I get
\[3 < 4x.\]
Dividing by 4, I get
\[\frac{3}{4} < x.\]
This means that x must be greater than 3/4 and less than 1 to satisfy the inequality.
So this is part of the solution set.
Now, if x is greater than 1, then the expression inside the absolute value is positive, so I can rewrite the inequality as
\[\frac{2x - 1}{x - 1} > 2.\]
To solve this, I can multiply both sides by x - 1, which is positive, and keep the same inequality sign.
I get
\[2x - 1 > 2x - 2.\]
Simplifying, I get
\[1 > 0.\]
This is always true, regardless of the value of x. So any x that is greater than 1 satisfies the inequality.
This is also part of the solution set.
Therefore, the solution set is the union of the two intervals
\[\left( \frac{3}{4}, 1 \right) \cup (1, \infty).\]
I can write this using set notation as
\[\{x : x > \frac{3}{4} \text{ and } x \neq 1\}.\]