What is the solution set for the inequality (2x - 5)/(x - 1) ≤ 1?

To solve the inequality ((2x - 5)/(x - 1) ≤ 1), we start by manipulating the inequality to isolate terms on one side.

First, subtract 1 from both sides:

[

\frac{2x - 5}{x - 1} - 1 ≤ 0

]

This can be rewritten with a common denominator:

[

\frac{2x - 5 - (x - 1)}{x - 1} ≤ 0

]

Simplifying the numerator gives:

[

\frac{2x - 5 - x + 1}{x - 1} ≤ 0

]

[

\frac{x - 4}{x - 1} ≤ 0

]

Next, we find the critical points of the inequality by setting the numerator and denominator to zero:

1. The numerator (x - 4 = 0) gives us (x = 4).

2. The denominator (x - 1 = 0) gives (x = 1).

These points are important because they help us determine the sign of the expression (\frac{x - 4}{x -