Find all xx with 7/(x−1) ≤ 2/(x−8)

We are tasked with solving the inequality:

\[
\frac{7}{x-1} \leq \frac{2}{x-8}
\]

### Step 1: Clear denominators
To eliminate the denominators, we multiply both sides of the inequality by \((x-1)(x-8)\). However, we need to consider the signs of \((x-1)\) and \((x-8)\) because multiplying by a negative number reverses the inequality.

We will solve the inequality in different intervals based on the critical points \(x = 1\) and \(x = 8\).

### Step 2: Solve in different intervals

1. **Interval 1: \(x < 1\)**

   Both \(x - 1 < 0\) and \(x - 8 < 0\), so the product \((x-1)(x-8) > 0\). The inequality remains unchanged:
   
   \[
   7(x-8) \leq 2(x-1)
   \]

   Simplify:
   
   \[
   7x - 56 \leq 2x - 2
   \]
   
   Subtract \(2x\) from both sides:
   
   \[
   5x - 56 \leq -2
   \]
   
   Add 56 to both sides:
   
   \[
   5x \leq 54
   \]
   
   Divide by 5:
   
   \[
   x \leq \frac{54}{5} = 10.8
   \]

   So, in this interval, the solution is \(x \leq 10.8\), but we must also have \(x < 1\) to stay within the interval. Therefore, there is no solution in the interval \(x < 1\).

2. **Interval 2: \(1 < x < 8\)**

   Here, \(x - 1 > 0\) and \(x - 8 < 0\), so \((x-1)(x-8) < 0\). We reverse the inequality:

   \[
   7(x-8) \geq 2(x-1)
   \]
   
   Simplify:
   
   \[
   7x - 56 \geq 2x - 2
   \]
   
   Subtract \(2x\) from both sides:
   
   \[
   5x - 56 \geq -2
   \]
   
   Add 56 to both sides:
   
   \[
   5x \geq 54
   \]
   
   Divide by 5:
   
   \[
   x \geq 10.8
   \]
   
   However, this is not possible in the interval \(1 < x < 8\), so there is no solution in this interval.

3. **Interval 3: \(x > 8\)**

   Both \(x - 1 > 0\) and \(x - 8 > 0\), so the product \((x-1)(x-8) > 0\), and the inequality remains the same:

   \[
   7(x-8) \leq 2(x-1)
   \]

   Simplify:

   \[
   7x - 56 \leq 2x - 2
   \]

   Subtract \(2x\) from both sides:

   \[
   5x - 56 \leq -2
   \]

   Add 56 to both sides:

   \[
   5x \leq 54
   \]

   Divide by 5:

   \[
   x \leq 10.8
   \]

   So, in this interval, the solution is \(8 < x \leq 10.8\).

### Step 3: Combine the solutions

The only valid solution occurs in the interval:

\[
8 < x \leq 10.8
\]

Thus, the solution to the inequality is:

\[
x \in (8, 10.8]
\]

Would you like further details on any step, or do you have questions?

---

### Related Questions:
1. How do you solve inequalities with rational expressions in general?
2. What are the implications of multiplying inequalities by expressions involving variables?
3. Can you show how to graph the solution \(x \in (8, 10.8]\)?
4. How do you check the solution of a rational inequality by substitution?
5. What are some common pitfalls when solving inequalities like this?

**Tip:** Always remember to check for restrictions on the variable, especially when dealing with fractions and inequalities.

Explore the step-by-step solution to the inequality 7/(x−1) ≤ 2/(x−8). This problem involves solving rational inequalities by clearing denominators, analyzing different intervals, and applying sign reversal. Ideal for students in Grades 10-12.

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