Question 150538

{{{3y+x>4}}} Start with the given inequality.



{{{3y>4-x}}} Subtract {{{x}}} from both sides.



{{{y>(4-x)/(3)}}} Divide both sides by {{{3}}} to isolate {{{y}}}.



{{{y>(4)/3-(x)/(3)}}} Break up the fraction.



{{{y>-(1/3)x+4/3}}} Rearrange the terms.



So in order to plot {{{3y+x>4}}} or {{{y>-(1/3)x+4/3}}} (which is the same thing), we need to plot the equation {{{y=-(1/3)x+4/3}}} first. 



{{{ graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3) }}} Graph of {{{y=-(1/3)x+4/3}}}



Now plug in a test point (0,0) into the inequality {{{y>-(1/3)x+4/3}}}



{{{y>-(1/3)x+4/3}}} Start with the given inequality.



{{{0>-(1/3)*(0)+4/3}}} Plug in {{{x=0}}} and {{{y=0}}}



{{{0>4/3}}} Evaluate and simplify.



Since the inequality is false, this means that we shade the entire region that does <b>NOT</b> contain the point (0,0)



In other words, we simply shade the <b>entire</b> region that is above the line.



 {{{drawing( 500, 500, -10, 10, -10, 10,
    graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+0),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+0.666666666666667),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+1.33333333333333),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+2),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+2.66666666666667),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+3.33333333333333),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+4),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+4.66666666666667),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+5.33333333333333),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+6),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+6.66666666666667),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+7.33333333333333),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+8),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+8.66666666666667),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+9.33333333333333),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+10),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+10.6666666666667),
graph( 500, 500, -10, 10, -10, 10,-(1/3)x+4/3,-(1/3)x+4/3+11.3333333333333)

 )}}} Graph of {{{y>-(1/3)x+4/3}}} (which is also {{{3y+x>4}}} )with the shaded region in green



Note: Since the inequality has a greater than sign ">", this means that the red boundary line should be dotted since we are not including this line with the solution set.