Solving Systems of Inequalities

A system of inequalities can be solved using Desmos.  When a system of inequalities is graphed, the region where the shaded portions overlap is the solution to the system.   Consider the following:

If the system of inequalities (y≥2x+2) and (y>3x−1) is graphed in the xy-plane above, which quadrant contains no solutions to the system? 

A) Quadrant I
B) Quadrant II
C) Quadrant III
D) Quadrant IV

When graphed, you should see the common solutions appear only in quadrants I, II, and II.

There is a variant of systems of inequalities to watch out for – when both equations do not have both variables.  The following exemplifies this:


Which of the following consists of the y-coordinates of all points that satisfy the system of inequalities above?

A) (y>7)
B) (y > 4)
C) (y>frac53)
D) (y<3)

In this case, we need to figure out how to compare these two inequalities as substitution and elimination aren’t viable.  Notice that both equations share a “3x”.   This is our clue for how to approach.  By modifying the second inequality slightly, we can make equivalent expressions.


Just like linear equations, we always do the same thing to both sides.  In this case, adding 2 gets us to (3x+2>7).  We can then use the transitive property of inequalities to say that if (3x+2) must be greater than 7, y must also be greater than 7.