Answer
See below:
Work Step by Step
Graph each inequality on the same plane.
Step 1: Convert the inequality to an equation by replacing inequality by = sign.
Equation 1: \[3x-y\le 6\]
\[3x-y=6\]
Equation 2: \[x+y\ge 2\]
\[x+y=2\]
Step 2: Graph the equation using intercept form or slope-intercept form.
Equation 1: \[3x-y\le 6\]
Make \[x=0\] to find the y-intercept,
\[\begin{align}
& 3x-y=6 \\
& 0-y=-6
\end{align}\]
Make \[y=0\] to find the x-intercept,
\[\begin{align}
& 3x-y=6 \\
& 3x-0=6 \\
& x=2
\end{align}\]
The line passes through \[\left( 0,-6 \right)\] and\[\left( 2,0 \right)\]. The line is solid since the inequality contains the ≤symbol.
Equation 2: \[x+y\ge 2\]
Make \[x=0\] to find the y-intercept,
\[\begin{align}
& x+y=2 \\
& 0+y=2 \\
& y=2
\end{align}\]
Make \[y=0\] to find the x-intercept,
\[\begin{align}
& x+y=2 \\
& x+0=2 \\
& x=2
\end{align}\]
The line passes through \[\left( 0,2 \right)\] and\[\left( 2,0 \right)\]. The line is solid since the inequality contains the ≥symbol.
Step 3: Choose a test point. The origin \[\left( 0,0 \right)\]be the test point. Substituting in the inequality
Equation 1: \[3x-y\le 6\]
\[0\le 6\]
The statement is true. The graph contains the test point.
Equation 2: \[x+y\ge 2\]
\[0\ge 2\]
The statement is false. The graph does not contain the test point.
