Answer:
D. [tex]y\geq 3x - 2\\x + 2y\leq 4[/tex]
Step-by-step explanation:
From the graph, we can conclude that,
1. The two lines are continuous lines and not broken lines. So, the inequality sign should be either [tex]\geq \textrm{ or }\leq[/tex].
2. The points on the lines of the shaded region are also included in the solution.
The only option that matches with the above conditions is option D. So, option D is the correct answer.
Let us verify it.
Now, let us consider a point that is inside the shaded region and also on any one line. Let us take [tex](0,2)[/tex].
Plug in 0 for x and 2 for y in each of the options and check which inequality holds true.
Option A:
[tex]y < 3x-2\\ 2 < 3(0)-2\\2<-2\\\\x + 2y \geq 4\\0+2(2)\geq 4\\4\geq 4[/tex]
So, inequality 1 is wrong as -2 is less than 2.
Option B:
[tex]y < 3x - 2\\ 2<3(0)-2\\2<-2\\\\x + 2y > 4\\0+2(2)>4\\4>4[/tex]
Both the inequalities are wrong.
Option C:
[tex]y > 3x - 2\\2>3(0)-2\\2>-2\\\\x + 2y < 4\\0+2(2)<4\\4<4[/tex]
Inequality 2 is wrong.
Option D:
[tex]y\geq 3x - 2\\2\geq 3(0)-2\\2\geq -2\\\\x + 2y\leq 4\\0+2(2)\leq 4\\4\leq 4[/tex]
Here, both inequalities are correct.
So, option D is the correct answer.
