We need to find two points of the boundary line for the following inequality
[tex]2(y+4)<-x[/tex]in order to do this, first we will replace the sign of less than ( < ) by the sign of equal to ( = ), and then we replace x = 0 and then y = 0 to find the points.
1. replace the sign of less than ( < ) by the sign of equal to ( = )
[tex]2(y+4)=-x[/tex]2. Simpify
[tex]2y+8=-x[/tex]3. replace x = 0 and simplify to find the 1st point
[tex]\begin{gathered} 2y+8=0 \\ 2y=-8 \\ y=-\frac{8}{2}=-4 \end{gathered}[/tex]So, our 1st point is (0,-4)
4. replace y = 0 and simplify to find the 2nd point
[tex]\begin{gathered} 2\cdot0+8=-x \\ 0+8=-x \\ x=-8 \end{gathered}[/tex]so, our 2nd point is (-8,0)
Now, to determine the shaded region we need to simplify the inequality in the following way:
1. write the inequality
[tex]2(y+4)<-x[/tex]2. eliminate parentheses
[tex]2y+8<-x[/tex]3. substract 8 on both sides and simplify
[tex]\begin{gathered} 2y+8-8<-x-8 \\ 2y<-x-8 \end{gathered}[/tex]4. divide 2 on both sides and simplify
[tex]\begin{gathered} \frac{2y}{2}<\frac{-x-8}{2} \\ y<-\frac{1}{2}x-4 \end{gathered}[/tex]Since the sign of the inequality is less than, this indicates that the shaded region is the lower region.
Summary
The answers are the following:
1) (0,-4) (-8,0)
2) lower region
