Question 28678
First, plot or sketch the first 2 straight lines:


{{{graph(300,300,-3,10,-2,6,(-3/4)x+3, (-1/3)x+2)}}}


{{{3x+4y <= 12}}} is the brown line, also quoted as {{{y <= (-3/4)x + 3}}}
{{{x+3y <= 6}}} is the green line, also quoted as {{{y <= (-1/3)x + 2}}}


So, for the first equation... pick a point that is NOT on the line. And pick a nice easy point ie one with small numbers...i choose the origin (0,0).


y and  (-3/4)x + 3
--> 0 and (-3/4)(0) + 3
--> 0 and 0 + 3
--> 0 and 3
--> zero is less than 3
so {{{y <= (-3/4)x + 3}}} is the region below the line... SHADE EVERYTHING ABOVE THE LINE, as this is NOT wanted.


Repeat for the second line, and pick a point, again i pick (0,0):
y and (-1/3)x + 2
--> 0 and (-1/3)(0) + 2
--> 0 and 0+2
--> 0 and 2
--> zero is LESS than 2
so {{{y <= (-1/3)x + 2}}} is the region below that line... SHADE EVERYTHING ABOVE THE LINE, as this is NOT wanted.



And now, the third line, {{{x>=0}}}. Well x=0 is the y-axis. And we want every point/place on the graph that is greater or equal to that line...this is everything to the right of the y-axis. SHADE EVERYTHING TO THE LEFT OF THE y-axis, as this is NOT wanted.



And now, the last line, {{{y >= 0}}}. Well, y=0 is the x-axis. And we want every point/place on the graph that is greater or equal to that line... this is everything above the x-axis. SHADE EVERYTHING BELOW THE x-axis, as this is NOT wanted.


What region is left clear? is is the little 4-sided shape in the middle, bounded by the x- and y-axis and the 2 straight lines forming a ^ shape.


jon.