Answer
The graph of the solution set is shown below.
Work Step by Step
The given system of inequalities contains two linear inequalities. We will solve each of them, then intersect the two solution sets.
For $x+2y\geq2$
Replace the inequality symbol by $=$ and graph the linear equation.
$\Rightarrow x+2y=2$.
Plug $y=0$ into the linear equation.
$\Rightarrow x+2(0)=2$
$\Rightarrow x=2$
The $x-$ intercept is $2$, so the line passes through $A=(2,0)$.
Plug $x=0$ into the linear equation.
$\Rightarrow (0)+2y=2$
$\Rightarrow 2y=2$
Divide both sides by $2$.
$\Rightarrow \frac{2y}{2}=\frac{2}{2}$
Simplify.
$\Rightarrow y=1$
The $y-$ intercept is $1$, so the line passes through $B=(0,1)$.
Draw a solid line through these two points because equality is included.
Choose a test point.
Let the test point is $C=(0,0)$.
Substitute the test point into given inequality.
$\Rightarrow 0+2(0)\geq2$.
Simplify.
$\Rightarrow 0\geq2$. The statement is false.
Shade the half-plane not containing the test point.
For $x-y\geq-4$
Replace the inequality symbol by $=$ and graph the linear equation.
$\Rightarrow x-y=-4$.
Plug $y=0$ into the linear equation.
$\Rightarrow x-(0)=-4$
$\Rightarrow x=-4$
The $x-$ intercept is $-4$, so the line passes through $D=(-4,0)$.
Plug $x=0$ into the linear equation.
$\Rightarrow (0)-y=-4$
$\Rightarrow -y=-4$
Mulitply both sides by $-1$.
$\Rightarrow -1(-y)=-1(-4)$
Simplify.
$\Rightarrow y=4$
The $y-$ intercept is $4$, so the line passes through $E=(0,4)$.
Draw a solid line through these two points because equality is included.
Choose a test point.
Let the test point is $F=(0,0)$.
Substitute the test point into given inequality.
$\Rightarrow 0-0\geq-4$.
Simplify.
$\Rightarrow 0\geq-4$. The statement is true.
Shade the half-plane containing the test point.
The intersection (overlap) of the two half-planes is the solution set.
The combined graph is shown below.
