Answer
The solution graph is shown below.
Work Step by Step
The given system of inequalities is
$\left\{\begin{matrix}
3x& +6y&\leq&6\\
2x& +y & \leq&8
\end{matrix}\right.$
First we graph both inequalities.
For $3x +6y\leq6$
Replace the inequality symbol with $=$ and graph the linear equation.
$3x +6y=6$
Plug $y=0$ for the $x−$intercept.
$\Rightarrow 3x +6(0)=6$
$\Rightarrow 3x=6$
Divide both sides by $3$.
$\Rightarrow \frac{3x}{3}=\frac{6}{3}$
Simplify.
$\Rightarrow x=2$
The $x−$intercept is $2$, so the line passes through $A=(2,0)$.
Plug $x=0$ for the $y−$intercept.
$\Rightarrow 3(0) +6y=6$
Simplify.
$\Rightarrow 6y=6$
Divide both sides by $6$.
$\Rightarrow \frac{6y}{6}=\frac{6}{6}$
Simplify.
$\Rightarrow y=1$
The $y−$intercept is $1$, so the line passes through $B=(0,1)$.
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.
For $2x +y\leq8$
Replace the inequality symbol by $=$ and graph the linear equation.
$2x +y=8$
Plug $y=0$ for the $x−$intercept.
$\Rightarrow 2x +(0)=8$
$\Rightarrow 2x=8$
Divide both sides by $2$.
$\Rightarrow \frac{2x}{2}=\frac{8}{2}$
Simplify.
$\Rightarrow x=4$
The $x−$intercept is $4$, so the line passes through $C=(4,0)$.
Plug $x=0$ for the $y−$intercept.
$\Rightarrow 2(0) +y=8$
Simplify.
$\Rightarrow y=8$
The $y−$intercept is $8$, so the line passes through $D=(0,8)$.
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 $E=(\frac{14}{3},\frac{-4}{3})$ is in the solution set because it satisfies both inequalities.
The combined graph is shown below.
