The shaded region in the following figure is the solution set of the inequations
The shaded region in the following figure is the solution set of the inequations
$\begin{aligned} & x+2 y \leq 6,5 x+3 y \geq 15, x \leq 7, y \leq 6, x \text {, } \\ & y \geq 0\end{aligned}$
$\begin{aligned} & x+2 y \geq 6,5 x+3 y \geq 15, x \leq 7, y \leq 6, x \\ & y \geq 0\end{aligned}$
$\begin{aligned} & x+2 y \geq 6,5 x+3 y \leq 15, x \geq 7, y \leq 6, x, \\ & y \geq 0\end{aligned}$
$\begin{aligned} & x+2 y \leq 6,5 x+3 y \leq 15, x \leq 7, y \geq 6, x, \\ & y \geq 0\end{aligned}$
Solution
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):
$\text { (1) } \quad x+2 y \leq 6, \quad 5 x+3 y \geq 15, \quad x \leq 7, \quad y \leq 6, \quad x, y \geq 0$
