The given problem states:

Suppose $3 < a < 4$ and $4 < b < 5$. Find all possible values of the expression:

$\frac{a - b}{2a}$

Step-by-step solution:

1. Bounds for $a$ and $b$:

   • $a \in (3, 4)$
   • $b \in (4, 5)$

2. Expression to evaluate: $\frac{a - b}{2a}$

Evaluate for the extreme values:

• For $a = 3$ (approaching from the right) and $b = 5$ (approaching from the left):

  $\frac{3 - 5}{2(3)} = \frac{-2}{6} = -\frac{1}{3}$

• For $a = 4$ (approaching from the left) and $b = 4$ (approaching from the right):

  $\frac{4 - 4}{2(4)} = \frac{0}{8} = 0$

Thus, the expression $\frac{a - b}{2a}$ can range between $-\frac{1}{3}$ and $0$.

Final Answer:

The possible values of the expression $\frac{a - b}{2a}$ range between $-\frac{1}{3}$ and $0$, i.e.,

$-\frac{1}{3} \leq \frac{a - b}{2a} \leq 0$

Would you like more details or have any questions about the solution? Here are five related questions to explore:

1. How would the range of the expression change if $a$ and $b$ were constrained differently?
2. What happens if the denominator involves $b$ instead of $a$?
3. Can you find the maximum and minimum values of $\frac{a - b}{2a}$ using calculus techniques?
4. How does the behavior of the function change if we add another variable or condition?
5. What would the expression look like if we inverted it (i.e., $\frac{2a}{a - b}$)?

Tip: When working with inequalities involving variables, always test the extreme values to understand the range of possible outcomes.