Answer
See below:
Work Step by Step
Consider the provided inequalities:
\[x+2y\le 4\]and\[y\ge x-3\]
Replace each inequality symbol with an equal sign:
\[x+2y=4\]and\[y=x-3\]
Now, draw the graph of the equation\[x+2y=4\].
Put \[y=0\] for x-intercept and \[x=0\] for y-intercept in the equation \[x+2y=4\].
So, the x-intercept is 4 and y-intercept is 2.
Therefore, the line is passing through \[\left( 4,0 \right)\]and \[\left( 0,2 \right)\].
Now, consider a test point \[\left( 0,0 \right)\], which lies in the half-plane.
Substitute \[x=0\] and \[y=0\]in \[x+2y\le 4\].
\[\begin{align}
& 0+2\cdot 0\le 4 \\
& 0\le 4
\end{align}\]
Since \[\left( 0,0 \right)\]satisfies the above inequality \[x+2y\le 4\], the test point \[\left( 0,0 \right)\] is part of the solution set.
All the points are on the same side of the line \[x+2y=4\],as the points\[\left( 0,0 \right)\] are the members of the solution set.
Since,\[x+2y\le 4\] contains an equal sign, the line should be solid.
Now, draw the graph of the equation\[y=x-3\].
Put \[y=0\] for x-intercept and \[x=0\] for y-intercept in the equation,\[y=x-3\].
So, the x-intercept is 3 and y-intercept is \[-3\].
Therefore, the line is passing through \[\left( 3,0 \right)\]and \[\left( 0,-3 \right)\].
Now, consider a test point \[\left( 0,0 \right)\], which lies in the half-plane.
Substitute \[x=0\] and \[y=0\]in \[y\ge x-3\].
\[\begin{align}
& 0\ge 0-3 \\
& 0\ge -3
\end{align}\]
Since \[\left( 0,0 \right)\]satisfies the above inequality \[y\ge x-3\], the test point \[\left( 0,0 \right)\] is part of the solution set.
All the points are on the same side of the line \[y=x-3\]as the point \[\left( 0,0 \right)\] are members of the solution set.
Since,\[y\ge x-3\] contains an equal sign, the line should be solid.
Therefore, the graph of the linear inequality\[x+2y\le 4\] and \[y\ge x-3\] is provided below:
