Question 148204
{{{6+3y<4(3-x)}}} Start with the given inequality.



{{{6+3y<12-4x}}} Distribute.



{{{3y<12-4x-6}}} Subtract {{{6}}} from both sides.



{{{3y<-4x+6}}} Combine like terms on the right side.



{{{y<(-4x+6)/(3)}}} Divide both sides by {{{3}}} to isolate {{{y}}}. 



{{{y<(-4x)/(3)+(6)/(3)}}} Break up the fraction.



{{{y<-(4/3)x+2}}} Reduce


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Answer:


So the answer is {{{y<-(4/3)x+2}}} 




Now let's graph the equation {{{y=-(4/3)x+2}}} (replace the inequality sign with an equals sign)



{{{ graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2) }}} Graph of {{{y=-(4/3)x+2}}}



Now plug in a test point (0,0) into the inequality {{{y<-(4/3)x+2}}}



{{{y<-(4/3)x+2}}} Start with the given inequality.



{{{0<-(4/3)*(0)+2}}} Plug in {{{x=0}}} and {{{y=0}}}



{{{0<2}}} Evaluate and simplify.



Since the inequality is true, this means that we shade the entire region that contains the point (0,0)



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



 {{{drawing( 500, 500, -10, 10, -10, 10,
    graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-0),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-2.66666666666667),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-5.33333333333333),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-8),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-10.6666666666667),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-13.3333333333333),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-16),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-18.6666666666667),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-21.3333333333333),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-24),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-26.6666666666667),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-29.3333333333333),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-32),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-34.6666666666667),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-37.3333333333333),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-40),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-42.6666666666667),
graph( 500, 500, -10, 10, -10, 10,-(4/3)x+2,-(4/3)x+2-45.3333333333333)

 )}}} Graph of {{{y<-(4/3)x+2}}} with the shaded region in green




note: the boundary should be a dotted line.