Solve the system of inequalities using shading

Question

y=> 2x^2+2x-3

y=< 1/3x+6

Graph inequalities and solve by shading.

Answer 1

First graph the function y = 2x² + 2x - 3 (this will be the boundary line you will use in shading). Then graph the function y = 1/3x + 6 (this will also be the boundary line you will use in shading. Since both of these boundary lines involve equalities (either < or >) then the boundary lines are to be drawn as solid lines (if they were strictly inequalities (< or >) then you would use a dotted line).

Then you shade according to the details of each function. for y > 2x² + 2x - 3 you will shade above the boundary because the y value is greater (or equal to) the function boundary (shown in red hatching below). For y < 1/3x + 6 you shade below the boundary line (because the y values must be less than (or equal to) the function boundary (shown in blue hatching below). The solution is the portion with both shadings. Like this:

The solution is in the shaded bluish violet section between the two curves.