We have to graph the solution set for the inequality:
[tex]2x\le-3y+9[/tex]The first step is to graph the function that divides the solution region from the other region. This line correspond to the equality within this inequality:
[tex]2x=-3y+9[/tex]If we rearrange it we can find two points to graph it:
[tex]\begin{gathered} 2x=-3y+9 \\ 2x+3y=9 \end{gathered}[/tex]When x=0, then y is:
[tex]\begin{gathered} 2\cdot0+3y=9 \\ y=\frac{9}{3} \\ y=3 \end{gathered}[/tex]Then, the y-intercept is at y=3.
When y=0, then x is:
[tex]\begin{gathered} 2x+3\cdot0=9 \\ x=\frac{9}{2} \end{gathered}[/tex]Now we now that the x-intercept is at x=9/2.
We have two points from the line, so we can graph it as:
Now, we know the line that limits the solution region.
As the inequality includes the equal sign, we know that this limit is included in the solution region.
The only thing left is to find is if the solution region is above this line or if it is below.
One easy way to test it is to select a point from one of the regions and replace (x,y) in the inequality: if the inequality stands true, then this point is in the solution region and we then now on which side the solution region is.
In this case, we can test with point (0,0) to make it easier:
[tex]\begin{gathered} (x,y)=(0,0)\Rightarrow2\cdot0\le-3\cdot0+9 \\ 0\le-0+9 \\ 0\le9\to\text{True} \end{gathered}[/tex]As the inequality is true for this point, we know that the solution region includes (0,0).
Then, we know that the solution region is below the line.
We then can graph it as:
