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.

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