Problem Analysis

We are given the ranges for two variables, X and Y:

• $3 \le X \le 5$
• $8 \le Y \le 11$

We need to determine the possible range for the fraction $\frac{X}{Y}$. To do this, we find the minimum and maximum possible values of the fraction.

Finding the Range of X/Y

The value of a fraction $\frac{X}{Y}$ is minimized when the numerator (X) is at its minimum and the denominator (Y) is at its maximum. Conversely, it is maximized when the numerator (X) is at its maximum and the denominator (Y) is at its minimum.

Minimum Value Calculation

Using the minimum value for X and the maximum value for Y:

• Minimum X = 3
• Maximum Y = 11
• Minimum value of $\frac{X}{Y}$ = $\frac{\min(X)}{\max(Y)}$ = $\frac{3}{11}$

Maximum Value Calculation

Using the maximum value for X and the minimum value for Y:

• Maximum X = 5
• Minimum Y = 8
• Maximum value of $\frac{X}{Y}$ = $\frac{\max(X)}{\min(Y)}$ = $\frac{5}{8}$

Determining the Correct Inequality

Combining the minimum and maximum values, we establish the range for $\frac{X}{Y}$:

$\frac{3}{11} \le \frac{X}{Y} \le \frac{5}{8}$

This inequality represents the correct range for the fraction $\frac{X}{Y}$ given the constraints on X and Y. Comparing this with the options provided, Option 2 accurately reflects this derived range.