Outside a home, there is a 4-key keypad with letters A, B, C, and D that can be used to open the garage if the correct four-letter code is entered. Each key may be used only once. How many codes are possible?

Hello there!

we have the point (3,2) which means that x = 3 and y = 2

The systems of equations are:

Y <  - 2

Y ≥ (2/3)x - 4

and

Y > - 2

Y ≤ (2/3)x - 4

The points were the systems are true in the white areas, the colored areas are the excluded ones.

Now we need to put our point in both systems and see if the point is a solution or not,

In the first one, you can see that y needs to be less than -2, and in out point y is equal to 2, then the point (3,2) cant is a solution of the first system.

let's see the second system:

Y > - 2

Y ≤ (2/3)x - 4

valuate it in the point (3,2)

2 > -2

2 ≤ (2/3)*3 - 4 = - 2

this is also false.

Then the point (3,2) is not a solution for neither system, and you can see it in the graphs, in the first graph the point (3,2) is in the black area, and in the second one is in the red area.