Explanation
[tex]\begin{gathered} -2x+3y\ge9 \\ x\ge-5 \\ y<6 \end{gathered}[/tex]
Step 1
graph the inequality (1)
a) isolate y
[tex]\begin{gathered} -2x+3y\geqslant9 \\ add\text{ 2x in both sides} \\ -2x+3y+2x\geqslant9+2x \\ 3y\ge9+2x \\ divide\text{ both sides by 3} \\ \frac{3y}{3}\geqslant\frac{9}{3}+\frac{2x}{3} \\ y\ge\frac{2}{3}x+3 \end{gathered}[/tex]
b) now, change the symbol to make an equality and find 2 points from the line
[tex]\begin{gathered} y=\frac{2}{3}x+3 \\ i)\text{ for x=0} \\ y=\frac{2}{3}(0)+3 \\ \text{sp P1\lparen0,3\rparen} \\ \text{ii\rparen for x=3} \\ y=\frac{2}{3}(3)+3=5 \\ so\text{ P2\lparen3,5\rparen} \end{gathered}[/tex]
now, draw a solid line that passes troguth those point
(0,3) and (3,5)
[tex]y\geqslant\frac{2}{3}x+3\Rightarrow y=\frac{2}{3}x+3\text{\lparen solid line\rparen}[/tex]
as we need the values greater or equatl thatn the function, we need to shade the area over the line
Step 2
graph the inequality (2)
[tex]x\ge-5[/tex]
this inequality represents the numbers greater or equal than -5 ( for x), so to graph the inequality:
a) draw an vertical line at x=-5, and due to we are looking for the values greater or equal than -5 we need to use a solid line and shade the area to the rigth of the line
Step 3
finally, the inequality 3
[tex]y<6[/tex]
this inequality represents all the y values smaller than 6, so we need to draw a horizontal line at y=6 and shade the area below the line
Step 4
finally, the solution is the intersection of the areas
I hope this helps you
