suppose 3<a<4 and 4<b<5. Find all possible values of each expression. (a-b)/2a

We are tasked with finding the possible range of values for the expression:

\[
\frac{a - b}{2a}
\]

where \( 3 < a < 4 \) and \( 4 < b < 5 \).

### Step 1: Analyzing the bounds for \( a - b \)
- \( a \) ranges between 3 and 4, so the smallest value for \( a \) is slightly greater than 3, and the largest value for \( a \) is slightly less than 4.
- \( b \) ranges between 4 and 5, so the smallest value for \( b \) is slightly greater than 4, and the largest value for \( b \) is slightly less than 5.

Thus, for \( a - b \):
- The minimum value of \( a - b \) occurs when \( a \) is near 3 and \( b \) is near 5. In that case:
  \[
  a - b \approx 3 - 5 = -2
  \]
- The maximum value of \( a - b \) occurs when \( a \) is near 4 and \( b \) is near 4. In that case:
  \[
  a - b \approx 4 - 4 = 0
  \]

So, \( a - b \) ranges between approximately \( -2 \) and 0.

### Step 2: Analyzing the bounds for \( 2a \)
- The minimum value of \( 2a \) occurs when \( a \) is near 3:
  \[
  2a \approx 2 \times 3 = 6
  \]
- The maximum value of \( 2a \) occurs when \( a \) is near 4:
  \[
  2a \approx 2 \times 4 = 8
  \]

Thus, \( 2a \) ranges between approximately 6 and 8.

### Step 3: Finding the possible values of \( \frac{a - b}{2a} \)
We now divide the range of \( a - b \) by the range of \( 2a \):
- The minimum value of \( \frac{a - b}{2a} \) occurs when \( a - b \) is at its minimum (\( -2 \)) and \( 2a \) is at its maximum (\( 8 \)):
  \[
  \frac{a - b}{2a} \approx \frac{-2}{8} = -\frac{1}{4}
  \]
- The maximum value of \( \frac{a - b}{2a} \) occurs when \( a - b \) is at its maximum (0) and \( 2a \) is at its minimum (6):
  \[
  \frac{a - b}{2a} \approx \frac{0}{6} = 0
  \]

### Conclusion:
The possible values of \( \frac{a - b}{2a} \) range between \( -\frac{1}{4} \) and 0. Therefore, the solution is:

\[
-\frac{1}{4} \leq \frac{a - b}{2a} \leq 0
\]

Would you like more details or have any questions? Here are five related questions to consider:

1. What happens to the value of \( \frac{a - b}{2a} \) as \( a \) approaches 4 and \( b \) approaches 5?
2. How would the range of \( \frac{a - b}{2a} \) change if \( a \) and \( b \) had different bounds?
3. What is the significance of the bounds on the numerator in relation to the denominator?
4. Can you generalize the solution for different ranges of \( a \) and \( b \)?
5. How does the range of \( a \) influence the magnitude of the overall expression?

**Tip**: When dealing with inequalities involving expressions, carefully analyze the maximum and minimum values of each part separately to ensure an accurate range.

Discover how to find the range of possible values for the expression (a-b)/2a, where 3 < a < 4 and 4 < b < 5. This solution covers key algebraic concepts, including working with inequalities and solving fractional expressions. Suitable for students in Grades 9-11.

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