Shade the region in the xy plane that is described by the inequalities 3x-y-7< 0 and x+5y+3> =0

To solve inequalities algebraically, first, you have to graph the lines disregarding the inequalities for a while.

For line 3x-y-7 <0, let's disregard < first and change it to =, such that 3x-y-7=0. Rearranging, y=3x-7. If you plot this equation, that would be the blue line in the equation. To know which of side of the line is the solution, you substitute a random point into the equality. For example, let's use point (5,20).

3x-y-7<0
3(5)-20-7<0
-12<0, this is true. Therefore, all the space to the left of the blue line is a solution (blue region).

We do the same for the other equation (orange line). Let's use the same point (5,20) to test.

x+5y+3≥0
5+5(20)+3≥0
108≥0, this is true, Therefore, everything above the orange line is a solution (orange region).

The overlapped area of the two shaded regions is presented as green in the picture. This is the exact solution of this system of linear equations.