The output of y=[ [ x ] ] is the greatest integer less than or equal to x.
Practice makes perfect
Let's start by graphing the boundary line for the given inequality.
Inequality & Boundary Line
y <[ [ x+2 ] ] & y = [ [ x+2 ] ]
To graph it, we will first consider the parent function y=[ [ x ] ], whose output is the greatest integer less than or equal to x. For example, [ [ 2.1 ] ] =2, [ [ 2.9 ] ] =2, and [ [ -1.7 ] ] =-2.
Let's now graph y=[ [ x+2 ] ]. This means we should move each segment of the function 2 units up.
Now, we need to test which region is the one that satisfies the inequality. Let's use (0,0) as our test point. If it produces a true statement, we will shade the region that contains (0,0). If not, we will shade the other region.
Since 0 is less than 2, we will shade the region which contains the point (0,0). Note that the inequality is strict, so the horizontal segments will be dashed.
Finally, we need to determine whether the vertical boundaries will be solid or dashed. To do so, we will use (1,2.5) to test the left boundaries. Let's do it!
