Question

100 POINTS!!! Graph each boundary line on the same graph

Answer 1

Answer: see image attached

y<x/2+6 is the yellowish area (with dashed boundary as that does not belong to the solution space)

x+3y>=12 is rearranged as y>=-x/3+4 and solutions correspond to the bluish shaded area (with solid line, due to the equal sign)

The solutions satisfying both inequalities are in the overlap area of the two.

Answer 2

<, > - boundary line is dashed

≤, ≥ - boundary line is solid

$y < \dfrac{1}{2}x+6\to y=\dfrac{1}{2}x+6\\\\for\ x=0\to y=\dfrac{1}{2}(0)+6=0+6=6\to(0,\ 6)\\\\for\ x=-6\to y=\dfrac{1}{2}(-6)+6=-3+6=3\to (-6,\ 3)\\\\\text{we shade below the line}\\\\x+3y\geq12\qquad|-x\\\\3y\geq-x+12\qquad|:3\\\\y\geq-\dfrac{1}{3}x+4\to y=-\dfrac{1}{3}x+4\\\\for\ x=0\to y=-\dfrac{1}{3}(0)+4=0+4=4\to(0,\ 4)\\\\for\ x=6\to y=-\dfrac{1}{3}(6)+4=-2+4=2\to(6,\ 2)\\\\\text{we shade above the line}$