Answer
(a)
Graph each inequality on the same plane.
Step 1: Convert the inequality to an equation by replacing inequality by = sign.
Equation 1: \[x\le 6\]
\[x=6\]
Equation 2: \[y\le 5\]
\[y=5\]
Equation 3: \[x+y\ge 2\]
\[x+y=2\]
Equation 1: \[x\le 6\]
The equation does not contain y. For any value of y, x is constant. The graph will be parallel to y-axis.
The line passes through \[\left\{ \ldots \left( 6,-1 \right),\left( 6,0 \right),\left( 6,1 \right)\ldots \right\}\]
The line is solid, since the inequality contains \[\le \] symbol.
Equation 2: \[y\le 5\]
The equation does not contain x. For any value of x, y is constant. The graph will be parallel to x-axis.
The line passes through \[\left\{ \ldots \left( -1,5 \right),\left( 0,5 \right),\left( 1,5 \right)\ldots \right\}\]
The line is solid, since the inequality contains \[\le \] symbol.
Equation 3: \[x+y\ge 2\]
Make \[x=0\] to find y-intercept;
\[\begin{align}
& x+y=2 \\
& 0+y=2 \\
& y=2
\end{align}\]
Make \[y=0\] to find 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 ≥ symbol.
Choose a test point for \[x+y\ge 2\]. The origin \[\left( 0,0 \right)\] be the test point. Substituting in the inequality:
\[0\ge 2\]
The statement is false. The graph does not contain the test point.
So, the graph of the system is as provided below:
Work Step by Step
