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To solve the inequality:

\[
\frac{x - 1}{x - 6} \leq 0
\]

we can break it down into steps by finding critical points, determining the sign of each interval, and then combining results based on where the inequality holds.

### Step 1: Find the Critical Points
The inequality \(\frac{x - 1}{x - 6} \leq 0\) involves two parts: the numerator and the denominator.

1. **Numerator**: \(x - 1 = 0 \Rightarrow x = 1\)
2. **Denominator**: \(x - 6 = 0 \Rightarrow x = 6\)

These values split the number line into intervals where the expression can change sign.

### Step 2: Set Up Intervals
The critical points \(x = 1\) and \(x = 6\) divide the number line into three intervals:
- \((- \infty, 1)\)
- \((1, 6)\)
- \((6, \infty)\)

### Step 3: Test Each Interval
We’ll test a point in each interval to determine whether \(\frac{x - 1}{x - 6}\) is positive or negative in that region.

1. **Interval \((- \infty, 1)\)**:
   - Choose \(x = 0\): \(\frac{0 - 1}{0 - 6} = \frac{-1}{-6} = \frac{1}{6} > 0\)

2. **Interval \((1, 6)\)**:
   - Choose \(x = 2\): \(\frac{2 - 1}{2 - 6} = \frac{1}{-4} = -\frac{1}{4} < 0\)

3. **Interval \((6, \infty)\)**:
   - Choose \(x = 7\): \(\frac{7 - 1}{7 - 6} = \frac{6}{1} = 6 > 0\)

### Step 4: Determine the Solution
We want the expression \(\frac{x - 1}{x - 6} \leq 0\). This is satisfied where the expression is negative or zero:
- In the interval \((1, 6)\), the expression is negative.
- At \(x = 1\), the expression is zero (since the numerator \(x - 1 = 0\) while the denominator is non-zero).
  
Thus, the solution in interval notation is:

\[
[1, 6)
\]

### Final Answer
The solution to the inequality is:
\[
[1, 6)
\]

Would you like further details or have any questions?

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Here are five related questions to expand your understanding:

1. How would the solution change if the inequality were \(\frac{x - 1}{x - 6} < 0\) instead of \(\leq\)?
2. What if the inequality were \(\frac{x - 1}{x - 6} \geq 0\)?
3. Can you solve \(\frac{x + 2}{x - 3} \geq 0\) using the same method?
4. How would the solution change if the denominator had a positive constant, like \(\frac{x - 1}{x + 6} \leq 0\)?
5. Why do we exclude \(x = 6\) from the interval solution?

**Tip:** Always check the sign of the inequality and handle cases where the denominator is zero, as it defines regions where the expression is undefined.

Solve the following inequality: (x - 1) / (x - 6) ≤ 0. Write your answer using interval notation.

Learn how to solve the rational inequality (x - 1) / (x - 6) ≤ 0 with step-by-step guidance. This solution involves sign analysis, interval notation, and critical points. Suitable for students in Grades 10-12.

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