graph inequalities

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graph inequalities

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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x+y>4 y+3x<6
|dw:1439648281026:dw|
Is that right, if so what would be the solution

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|dw:1439648545613:dw|
start by graphing the corresponding "lines" x+y = 4 y+3x = 6
|dw:1439648848102:dw|
you should get those lines, right ?
yes, so the solution the intrsection right?
the solution is just the intersection point if they are "equalities" but we have inequalities, so no
the solution is the shaded region
lets do the shading first x+y>4 y > -x + 4 so shade the above part for red line : |dw:1439649623303:dw|
do the shading for the other inequality also y+3x<6 y < -3x+6 so shade the bottom part of blue line : |dw:1439649749122:dw|
the solution is the common region of those two shadings
can we express that mathematically?
sure, you can see that \(x<1\) for the common region, yes ?
and every point in that region is above red line and below blue line : |dw:1439651204202:dw|