Question 87378

In order to graph {{{3x-4y<12}}}, we need to graph the <b>equation</b> {{{3x-4y=12}}} (just replace the inequality sign with an equal sign).
So lets graph the line {{{3x-4y=12}}} (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)


{{{ graph( 500, 500, -20, 20, -20, 20, (3/4)x-3) }}} graph of {{{3x-4y=12}}} 

Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality {{{3x-4y<12}}}


Substitute (0,0) into the inequality

{{{3(0)-4(0)<12}}} Plug in {{{x=0}}} and {{{y=0}}}

{{{0<12}}} Simplify

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

{{{drawing( 500, 500, -20, 20, -20, 20,
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+2),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+4),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+6),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+8),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+10),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+12),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+14),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+16),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+18),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+20),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+22),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+24),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+26),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+28),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+30),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+32),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+34),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+36),
graph(  500, 500, -20, 20, -20, 20,(3/4)x-3,(3/4)x-3+38))}}} Graph of {{{3x-4y<12}}} with the boundary (which is the line {{{3x-4y=12}}} in red) and the shaded region (in green) 
(note: since the inequality contains a less-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)