Begin by determining the boundary lines of the inequalities.
Graph:
Solution: No solution
Practice makes perfect
To graph the given system of inequalities, we will first determine the boundary lines of the inequalities that form the system.
y≥ |x-2|+4 & (I) y ≤ [ [ x ] ]-3 & (II)
Let's begin by determining the boundary line of Inequality I.
&Inequality I &&Boundary Line I
&y≥ |x-2|+4 && y= |x-2|+4
The graph of the boundary line is the graph of the parent function y=|x| translated 2 units to the right and 4 units up. Notice that because the inequality is non-strict, the line will be solid. Let's graph it!
Next, we can test the point (0,0) to decide which region we should shade.
The point did not satisfy the inequality, so we will shade the region that does not contain the test point.
We will determine the boundary line of Inequality II proceeding in the same way.
&Inequality II &&Boundary Line II
&y≤ [ [ x ] ]-3 && y= [ [ x ] ]-3
The boundary line is a special step function called greatest integer function. The graph of it is the graph of the parent function y=[ [ x ] ] translated 3 units to the right. It is also solid because the inequality is non-strict. Let's graph it!
The points with y-coordinates less than or equal to [ [ x ] ]-3 are included in the solution set of the inequality. Therefore, we will shade the region below the boundary line.
Finally, we will graph the inequalities on the same coordinate plane to find the solution set of the system.
Since the inequalities have no shaded region in common, the system has no solution.
