Answer
2x - 3y > -12 (in red ink)
x + y ≥ -2 (in black ink)
The solution region is the region that the two shaded regions have in common.
Step-by-step explanation
When plotting the graph of linear inequality equations, the first step is to first plot the graph of the straight line normally, using intercepts to generate two points on the linear graph.
If the inequality sign is (< or >), then the line drawn will be a broken line.
If the inequality sign is (≤ or ≥), then the line drawn is an unbroken one.
Step 1
For this question, we easily see that the first inequality will have a broken line and the second one will have an unbroken line.
To plot each of the lines, we will use intercepts to obtain the coordinates of two points on each line
Recall, we will first plot the lines like they are equations of a straight line.
To plot the graph
2x - 3y = -12
when x = 0,
2(0) - 3y = -12
-3y = -12
Divide both sides by -3
(-3y/-3) = (-12/-3)
y = 4
First point on the line is (0, 4)
when y = 0
2x - 3(0) = -12
2x = -12
Divide both sides by 2
(2x/2) = (-12/2)
x = -6
Second point on the line is (-6, 0)
For the second line,
To plot the graph,
x + y = -2
when x = 0
0 + y = -2
y = -2
First point on the line is (0, -2)
when y = 0
x + 0 = -2
x = -2
Second point on the line is (-2, 0)
So, for the plotting, we connect the two points for each of the lines.
Step 2
The shaded region now depends on whether the inequality sign is facing y or not.
If the inequality sign is facing y, it means numbers above the line plotted are the wanted region and the upper part of the graph is shaded.
If the inequality sign is not facing y, it means numbers below the line plotted are the wanted region and the lower part of the graph is shaded.
2x - 3y > -12
Can be rewritten as
-3y > -2x - 12
Divide through by -3 (this changes the inequality sign)
y < (2x/3) + 4
Here, we see that the inequality sign is not facing y, hence the numbers below the broken line plotted are the shaded region (in red ink)
x + y ≥ -2
We can rewrite this as
y ≥ -x - 2
Here, we see that the the inequality sign is facing y, hence, the numbers above the unbroken line plotted are the shaded region (in black ink)
The graph of this system of inequalities is presented above under 'Answer'
Hope this Helps!!!