Question 161963
In order to graph {{{-12x+8y<56}}}, we need to graph the <b>equation</b> {{{-12x+8y=56}}} (just replace the inequality sign with an equal sign).
So lets graph the line {{{-12x+8y=56}}} 



{{{ graph( 500, 500, -20, 20, -20, 20, (3/2)x+7) }}} graph of {{{-12x+8y=56}}} 



Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality {{{-12x+8y<56}}} with the test point


Substitute (0,0) into the inequality



{{{-12(0)+8(0)<56}}} Plug in {{{x=0}}} and {{{y=0}}}



{{{0<56}}} Simplify






Since this inequality is true, we simply shade the entire region that contains (0,0). In other words, we shade the entire region that is below the line.



{{{drawing( 500, 500, -20, 20, -20, 20,
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-4),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-8),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-12),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-16),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-20),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-24),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-28),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-32),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-36),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-40),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-44),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-48),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-52),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-56),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-60),
graph(  500, 500, -20, 20, -20, 20,(3/2)x+7,(3/2)x+7+-64))}}} Graph of {{{-12x+8y<56}}} with the boundary (which is the line {{{-12x+8y=56}}} 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)