Question

Find the graph of the inequality y>-1/6x+1

Answer 1

what we do is, we simply draw the graph for its EQUALITY counterpart, namely
`\bf y=-\cfrac{1}{6}x+1`.

then we check the regions on both sides of the graph, in this case we can check say the point (0,0) for simplicity, and

`\bf y>-\cfrac{1}{6}x+1\qquad (0,0)~~ \begin{cases} x=0\\ y=0 \end{cases}\implies 0>-\cfrac{1}{6}(0)+1\implies 0>1`

now, is 0 > 1? is 0 really greater than 1? nope, is smaller, thus the area where the point (0, 0) is at is the false region, so that's the area we do NOT shade, thus the other region must be the true region and that's the one we shade.

the ">" sign means, that "y" is greater than, NOT equals to but greater, thus the values at the borderline are not included, and thus is a dashed line.

Check the picture below.