Answer
The solution graph is shown below.
Work Step by Step
The given system of inequalities is
$\left\{\begin{matrix}
x& -y&\geq&4\\
x& +y & \leq&6
\end{matrix}\right.$
First we graph both inequalities.
For $x -y\geq4$
Replace the inequality symbol with $=$ and graph the linear equation.
$x -y=4$
Plug $y=0$ for the $x−$intercept.
$\Rightarrow x -0=4$
$\Rightarrow x=4$
The $x−$intercept is $4$, so the line passes through $A=(4,0)$.
Plug $x=0$ for the $y−$intercept.
$\Rightarrow 0 -y=4$
Simplify.
$\Rightarrow -y=4$
Multiply both sides by $-1$.
$\Rightarrow -1(-y)=-1(4)$
Simplify.
$\Rightarrow y=-4$
The $y−$intercept is $-4$, so the line passes through $B=(0,-4)$.
Draw a solid straight line through these intercept points because equality is included.
Greater than symbol with negative sign of the coefficient of $y$ indicates that the lower part of the line is the solution set.
For $x +y\leq6$
Replace the inequality symbol with $=$ and graph the linear equation.
$x +y=6$
Plug $y=0$ for the $x−$intercept.
$\Rightarrow x +(0)=6$
$\Rightarrow x=6$
The $x−$intercept is $6$, so the line passes through $C=(6,0)$.
Plug $x=0$ for the $y−$intercept.
$\Rightarrow (0) +y=6$
Simplify.
$\Rightarrow y=6$
The $y−$intercept is $6$, so the line passes through $D=(0,6)$.
Draw a solid straight line through these intercept points because equality is included.
Less than symbol indicates that the lower part of the line is the solution set.
The solution set of the system of inequalities is graphed as the intersection (the overlap) of the two half-planes.
The close dot at point $E=(5,1)$ is in the solution set because it satisfy both inequalities.
The combined graph is shown below.
