The output of y=[ [ x ] ] is the greatest integer less than or equal to x. The graph will look like a step function.
Practice makes perfect
Let's start by graphing the boundary line for the given inequality.
Inequality & Boundary Line
y ≥ | [ [ x ] ] | & y = | [ [ x ] ] |
To graph this step function 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 ] ] |. Recall that the absolute value will not affect the positive outputs, but it will turn the negative outputs into positive ones.
Now, we need to test which region is the one that satisfies the inequality. Let's use (2.5,0) as our test point. If it produces a true statement, we will shade the region that contains (2.5,0). If not, we will shade the other region.
Since 0 is notgreater than or equal to 2, we will shade the region which does not contain the point (2.5,0). Note that the inequality is not strict, so the horizontal segments will be solid.
Finally, we need to determine whether the vertical boundaries will be solid or dashed. To do so, we will use (1,0.5) to test the right boundaries, and (0,0.5) to test the left boundaries. Let's start by testing (1,0.5).
Since the point (1,0.5) produced a false statement, all the right boundaries will be dashed. Let's now test the point (0,0.5) to determine whether the left boundaries are dashed or solid.
