The shaded region in the following figure is the solution set of the inequations

To determine which option corresponds to the shaded region in the figure, let's analyze each inequality represented in the answer choices.
Axes and the Region: The region is bounded by the axes \(x \geq 0\) and \(y \geq 0\), which are present in all options. The other inequalities will help define the upper bounds more clearly.
Inequalities Exploration:
Option (1):
\(x+2 y \leq 6\) (below the line)
\(5 x+3 y \geq 15\) (above the line)
\(x \leq 7\) (to the left of the vertical line)
\(y \leq 6\) (below the horizontal line)
Option (2):
\(x+2 y \geq 6\) (above the line)
\(5 x+3 y \geq 15\) (above the line)
Options continue in the same manner with \(x\) and \(y\).
Interpreting the Shaded Region:
Examine if the inequalities allow a bounded area in the first quadrant.
The acceptable area must lie below the line for \(x+2 y \leq 6\) and above the line for \(5 x+3 y \geq 15\).
Summary of Options:
Option (1) fulfills the conditions that keep the region bounded below \(y=6\), to the left of \(x=7\), and within the first quadrant.
Following the analysis, the correct answer corresponds to the region described in Option (1):
\(\text { (1) } \quad x+2 y \leq 6, \quad 5 x+3 y \geq 15, \quad x \leq 7, \quad y \leq 6, \quad\) \(x, y \geq 0\)