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We are tasked with solving the inequality:

\[
\frac{5}{2c - 9} > 1
\]

Let's go through the steps to solve this inequality.

### Step 1: Subtract 1 from both sides
We want to move everything to one side of the inequality to make it easier to solve:

\[
\frac{5}{2c - 9} - 1 > 0
\]

### Step 2: Find a common denominator
We rewrite \( 1 \) as a fraction with the denominator \( 2c - 9 \):

\[
\frac{5}{2c - 9} - \frac{2c - 9}{2c - 9} > 0
\]

This simplifies to:

\[
\frac{5 - (2c - 9)}{2c - 9} > 0
\]

### Step 3: Simplify the numerator
Simplifying the expression in the numerator:

\[
5 - (2c - 9) = 5 - 2c + 9 = 14 - 2c
\]

So, the inequality becomes:

\[
\frac{14 - 2c}{2c - 9} > 0
\]

### Step 4: Analyze the sign of the fraction
We need to determine when the fraction \( \frac{14 - 2c}{2c - 9} \) is positive. A rational function is positive when the numerator and denominator have the same sign (both positive or both negative).

- **Numerator**: \( 14 - 2c \)
  - Positive when \( 14 - 2c > 0 \Rightarrow c < 7 \)
  
- **Denominator**: \( 2c - 9 \)
  - Positive when \( 2c - 9 > 0 \Rightarrow c > \frac{9}{2} = 4.5 \)

### Step 5: Solve using sign analysis
We now combine these two conditions to determine when the fraction is positive.

1. The numerator \( 14 - 2c \) is positive when \( c < 7 \).
2. The denominator \( 2c - 9 \) is positive when \( c > 4.5 \).

Thus, the fraction is positive when \( 4.5 < c < 7 \).

### Step 6: Consider undefined points
The expression is undefined when the denominator is zero, i.e., \( 2c - 9 = 0 \). Solving this gives \( c = 4.5 \), which is a critical value where the function is undefined. Therefore, we exclude \( c = 4.5 \) from the solution set.

### Final Answer:
The solution set in interval notation is:

\[
(4.5, 7)
\]

Would you like further details or have any questions?

Here are five related questions to further explore this topic:

1. How do we handle inequalities with rational expressions that have more than one critical value?
2. What happens if the inequality sign is \( \geq \) instead of \( > \)?
3. Can you solve a similar inequality where the fraction is negative instead of positive?
4. What is the difference between solving linear inequalities and solving rational inequalities?
5. How would you graph the solution set for this inequality?

**Tip:** When solving rational inequalities, always remember to check for values that make the denominator zero, as these points make the function undefined and must be excluded from the solution set.

Solve the inequality: 5 / (2c - 9) > 1. Write the solution set in interval notation if possible.

Learn how to solve the inequality 5 / (2c - 9) > 1 with a step-by-step solution. This problem covers rational inequalities, including the use of sign analysis and simplifying expressions. Suitable for students in Grades 10-12.

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