I really do not understand this question, can anyone explain it for me?

x and y are integers so we want the lattice points that satisfy all three inequalities.

I've plotted the inequalities in different colors.  

y≥2 is the aqua area above the horizontal line y=2.  

x+y≤4 is the pink area below the line through (4,0) and (0,4)

y≤2x+2 is the violet are below the line through (-1,0) and (0,2)

They form a little triangle in the middle of the graph.  The only lattice points in the triangle are on the sides, none in the interior.  We list them

(0,2), (1,2), (1,3), (2,2)

They all satisfy y≥2

Checking x+y≤4, they all satisfy that

Checking y≤2x+2 they all satisfy that.

Good; let's check a couple of nearby points to make sure they fail.

(1,4) fails x+y≤4

(0,3) fails y≤2x+2

(1,1) fails y≥2

(0,2), (1,2), (1,3), (2,2)

Step-by-step explanation:

they can be put into equations where they are:

y=2

x+y=4

y=2x+2

≥greater than or equal to means that the shaded area is on top of the line

≤less than or equal to means the shaded area is under the line

∞ this is your sign