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$$\begin{gathered}y \leq 3 x+1 \\x-y>1\end{gathered}$$Which of the following ordered pairs $(x, y)$ satisfies the system of inequalities above?

SAT · Math · previous-year question

  1. A.$(-2,-1)$
  2. B.$(-1,3)$
  3. C.$(1,5)$
  4. D.$(2,-1)$correct

Answer

D. $(2,-1)$

Explanation

Choice $\mathbf{D}$ is correct. 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.$$\begin{aligned}& y \leq 3 x+1 \\& y<x-1\end{aligned}$$Since $3 x+1>x-1$ for $x>-1$ and $3 x+1 \leq x-1$ for $x \leq-1$, it follows that $y<x-1$ for $x>-1$ and $y$ $\leq 3 x+1$ for $x \leq-1$. Of the given choices, only $(2,-1)$ satisfies these conditions because $-1<2-1$ $=1$.Alternate approach: Substituting $(2,-1)$ into th

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