Answer: The correct option is
[tex](2)~y+2x\leq-8,\\\\y+5>4x,\\\\-2x+7y\leq7.[/tex]
Step-by-step explanation: We are given to select the correct set of inequalities that represents the graph shown.
From the shaded region, let us consider a point (-6, -2).
The set of inequalities that will be satisfied by the point (x, y) = (-6, -2) is the required answer.
(1) The given set of inequalities are
[tex]-y\leq2x+8,\\\\y-4x<-5,\\\\7y-7\leq2x.[/tex]
For (x, y) = (-6, -2), we have
[tex]-y\leq2x+8\\\\\Rightarrow -(-2)\leq2(-6)+8\\\\\Rightarrow 2\leq-4,[/tex] which is not possible.
So, option (1) is NOT correct.
(2) The given set of inequalities are
[tex]y+2x\leq-8,\\\\y+5>4x,\\\\-2x+7y\leq7.[/tex]
For (x, y) = (-6, -2), we have
[tex]y+2x\leq-8\\\\\Rightarrow -2+2(-6)\leq-8\\\\\Rightarrow -14\leq-8,[/tex]true,
[tex]y+5>4x\\\\\Rightarrow -2+5>4(-6)\\\\\Rightarrow 3>-24,[/tex] true,
[tex]-2x+7y\leq7\\\\\Rightarrow -2(-6)+7(-2)\leq7\\\\\Rightarrow -2\leq7,[/tex] true.
Since all the three inequalities are satisfied by the point (-6, -2), so option (2) is CORRECT.
(3) The given set of inequalities are
[tex]-y\geq2x+8,\\\\y-4x<-5,\\\\7y-7\geq2x.[/tex]
For (x, y) = (-6, -2), we have
[tex]-y\geq2x+8\\\\\Rightarrow -(-2)\geq2(-6)+8\\\\\Rightarrow 2\geq-4,[/tex] true,
[tex]y-4x<-5\\\\\Rightarrow -2-4(-6)<-5\\\\\Rightarrow 22<-5,[/tex] which is not possible.
So, option (3) is NOT correct.
(4) The given set of inequalities are
[tex]y+2x\geq-8,\\\\y+5>4x,\\\\-2x+7y\geq7.[/tex]
For (x, y) = (-6, -2), we have
[tex]-y+2x\geq-8\\\\\Rightarrow -(-2)+2(-6)\geq-8\\\\\Rightarrow -10\geq-8,[/tex] which is not possible.
So, option (4) is NOT correct.
Thus, option (2) is CORRECT.
