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The inequality you provided is:

\[
\frac{1}{6 - x} < \frac{1}{x - 1}
\]

Let's solve this step by step.

### Step 1: Find common restrictions
Before proceeding, we must determine where the expressions are undefined. This happens when the denominators are zero.

- For \(\frac{1}{6 - x}\), the denominator is zero when \(6 - x = 0\), so \(x = 6\).
- For \(\frac{1}{x - 1}\), the denominator is zero when \(x - 1 = 0\), so \(x = 1\).

Thus, \(x = 6\) and \(x = 1\) are values where the expression is undefined. We must exclude these points from the solution set.

### Step 2: Multiply both sides by \((6 - x)(x - 1)\)
To eliminate the denominators, multiply both sides by the product \((6 - x)(x - 1)\). Note that the sign of the inequality may flip depending on the values of \(x\).

So the inequality becomes:

\[
(6 - x)(x - 1) \cdot \frac{1}{6 - x} < (6 - x)(x - 1) \cdot \frac{1}{x - 1}
\]

Canceling the terms:

\[
x - 1 < 6 - x
\]

### Step 3: Solve the linear inequality
Now, solve the inequality \(x - 1 < 6 - x\).

First, add \(x\) to both sides:

\[
2x - 1 < 6
\]

Next, add 1 to both sides:

\[
2x < 7
\]

Finally, divide by 2:

\[
x < \frac{7}{2} \quad \text{or} \quad x < 3.5
\]

### Step 4: Consider the restrictions
From Step 1, we know that \(x \neq 1\) and \(x \neq 6\). Additionally, the solution must take into account the regions defined by these points, so we must now analyze the behavior of the inequality in different intervals, namely:

- \(x < 1\)
- \(1 < x < 3.5\)
- \(3.5 < x < 6\)
- \(x > 6\)

### Step 5: Interval testing
- For \(x < 1\), both denominators are positive (the signs of \(6 - x\) and \(x - 1\) are the same), so the inequality holds.
- For \(1 < x < 3.5\), the inequality holds as we solved earlier.
- For \(x > 3.5\), the inequality does not hold because the expression \(\frac{1}{6 - x}\) becomes larger than \(\frac{1}{x - 1}\).

### Final Solution:
The solution is \(x \in (1, 3.5)\).

Let me know if you need more clarification or further steps! 

Here are some related questions:
1. What is the method for solving rational inequalities?
2. How do you handle undefined values in inequalities?
3. Can inequalities have multiple intervals in their solution?
4. What happens when multiplying both sides of an inequality by a negative expression?
5. How can we graph the solution of an inequality on a number line?

**Tip:** Always test values in different intervals when solving rational inequalities to ensure that the solution satisfies all parts of the inequality.

Solve the inequality 1/(6 - x) < 1/(x - 1)

Learn how to solve the inequality 1/(6 - x) < 1/(x - 1) with a step-by-step solution. This example covers key algebraic concepts like rational inequalities, restrictions on variables, and the correct handling of signs when multiplying through inequalities. Suitable for students in Grades 9-12.

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