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

$y \leq x + 7$

$y \geq - 2 x - 1$

Which point $(x, y)$ is a solution to the given system of inequalities in the xy-plane?

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Explanation

Choice D is correct. A point $(x, y)$ is a solution to a system of inequalities in the xy-plane if substituting the x-coordinate and the y-coordinate of the point for $x$ and $y$ , respectively, in each inequality makes both of the inequalities true. Substituting the x-coordinate and the y-coordinate of choice D, $14$ and $0$ , for $x$ and $y$ , respectively, in the first inequality in the given system, $y \leq x + 7$ , yields $0 \leq 14 + 7$ , or $0 less than or equals 21$ , which is true. Substituting $14$ for $x$ and $0$ for $y$ in the second inequality in the given system, $y \geq - 2 x - 1$ , yields $0 \geq - 2 (14) - 1$ , or $0 \geq - 29$ , which is true. Therefore, the point $left parenthesis 14 comma 0 right parenthesis$ is a solution to the given system of inequalities in the xy-plane.

Choice A is incorrect. Substituting $negative 14$ for $x$ and $0$ for $y$ in the inequality $y \leq x + 7$ yields $0 \leq - 14 + 7$ , or $0 \leq - 7$ , which is not true.

Choice B is incorrect. Substituting $0$ for $x$ and $negative 14$ for $y$ in the inequality $y \geq - 2 x - 1$ yields $- 14 \geq - 2 (0) - 1$ , or $- 14 \geq - 1$ , which is not true.

Choice C is incorrect. Substituting $0$ for $x$ and $14$ for $y$ in the inequality $y \leq x + 7$ yields $14 \leq 0 + 7$ , or $14 less than or equals 7$ , which is not true.