Solution:
Any point (x,y) that is a solution to the given system of inequalities must satisfy both inequalities in the system. Since the second inequality in the system can be rewritten as y<x−1, the system is equivalent to the following system.
y≤3x+1
y<x−1Since 3x + 1 > x − 1 for x>−1 and 3x+1≤x−1 forx≤−1, it follows that y<x−1 for x>−1 and y≤3x+1 for x≤−1. Of the given choices, only (2,−1) satisfies these conditions because −1<2−1=1.
Alternate approach: Substituting (2,−1)into the first inequality gives −1≤3(2)+1, or −1≤7,which is a true statement. Substituting (2,−1) into the second inequality gives 2 − (−1) > 1, or 3 > 1, which is a true statement. Therefore, since (2,−1) satisfies both inequalities, it is a solution to the system.
Choice A is incorrect because substituting −2 for x and −1 for y in the first inequality gives −1≤3(−2)+1, or −1≤−5, which is false. Choice B is incorrect because substituting −1 for x and 3 for y in the first inequality gives 3≤3(−1)+1, or 3≤−2, which is false. Choice C is incorrect because substituting 1 for x and 5 for y in the first inequality gives 5≤3(1)+1, or 5≤4, which is false.
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