Answer: The boundary lines are correct, but there shouldn't be any shaded region
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Explanation:
The graph of [tex]y \le x-1[/tex] will have us graph the boundary line y = x-1, and then shade below this solid boundary line. The boundary line y = x-1 is shown as the bottom boundary line of that diagram.
The graph of [tex]y \ge x+3[/tex] involves the boundary line y = x+3, with shading above the boundary line. The graph of y = x+3 is shown as the top boundary line of that diagram.
The boundary lines are correctly graphed, but the shaded region is incorrect. Instead, there shouldn't be any shaded region. The two shaded regions mentioned in the previous paragraphs do not overlap at all, so we don't have any (x,y) point that is in both shaded regions simultaneously. Therefore, there are no solutions to this entire system.
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Here's another reason why there's no solution region.
For each given inequality, subtract x from both sides
So [tex]y \le x-1[/tex] turns into [tex]y-x \le -1[/tex], while [tex]y \ge x+3[/tex] turns into [tex]y-x \ge 3[/tex]
Next, let z = y-x
We can replace each copy of 'y-x' with 'z' and we end up with the two inequalities [tex]z \le -1[/tex] and [tex]z \ge 3[/tex]
However, no such number is both -1 or smaller, AND 3 or larger
So there is no solution to that system involving z. Consequently, there are no solutions to the original system of inequalities.
