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:

$$y>3x+2\3x>5$$

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.

$$3x(+2)>5(+2)$$

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.