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Algebra / Linear inequalities in one or two variables Difficulty: Medium

$Inequality 1: y is less than or equal to x.$

$Inequality 2: y is less than or equal to negative x.$

Which of the following ordered pairs $x comma y$ is a solution to the system of inequalities above?

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Explanation

Choice D is correct. The solutions to the given system of inequalities is the set of all ordered pairs $x comma y$ that satisfy both inequalities in the system. For an ordered pair to satisfy the inequality $y is less than or equal to x$ , the value of the ordered pair’s y-coordinate must be less than or equal to the value of the ordered pair’s x-coordinate. This is true of the ordered pair $0 comma negative 1$ , because $negative 1 is less than or equal to 0$ . To satisfy the inequality $y is less than or equal to negative x$ , the value of the ordered pair’s y-coordinate must be less than or equal to the value of the additive inverse of the ordered pair’s x-coordinate. This is also true of the ordered pair $0 comma negative 1$ . Because 0 is its own additive inverse, $negative 1 is less than or equal to the negative of 0$ is the same as $negative 1 is less than or equal to 0$ . Therefore, the ordered pair $0 comma negative 1$ is a solution to the given system of inequalities.

Choice A is incorrect. This ordered pair satisfies only the inequality $y is less than or equal to x$ in the given system, not both inequalities. Choice B incorrect. This ordered pair satisfies only the inequality $y is less than or equal to negative x$ in the system, but not both inequalities. Choice C is incorrect. This ordered pair satisfies neither inequality.