Answer:
Option 3.
Step-by-step explanation:
This system of inequalities contains 2 lines.
[tex]2x - y = 3[/tex] (i)
and
[tex]x + 2y = 4[/tex] (ii)
The line (i) is the one that intercepts the y-axis in [tex]y = -3[/tex] and the x-axis in [tex]x = \frac{3}{2}[/tex]. Therefore the line (i) is drawn with blue in the graph.
The inequality is:
[tex]2x - y\geq 3[/tex]
We can clear the variable-y:
[tex]2x -3\geq y\\\\y \leq 2x-3[/tex]
Then, the region that represents this inequality are all the values in which the y-axis is less than the function [tex]f(x) = 2x-3.[/tex]. That is, the entire area below the blue line.
Then, line (ii) is represented by the color orange. It cuts the y axis in [tex]y = 2[/tex] and the x axis in [tex]x = 4[/tex]. Therefore we can discard the first two options shown in the image because there the orange line cuts in [tex]y = 4[/tex].
The inequality for this line is:
[tex]x + 2y <4[/tex]
so:
[tex]y <\frac{4 -x}{2}[/tex]
Then, the region that represents this inequality are all values in which the y-axis is smaller than the function [tex]f(x) = \frac{4 -x}{2}[/tex]. That is, the entire area below the orange line.
Finally the region sought is the one below the orange line and at the same time below the blue line. That is, the third option
