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

$y greater than 14$

$4 x plus y less than 18$

The point $left parenthesis x comma 53 right parenthesis$ is a solution to the system of inequalities in the xy-plane. Which of the following could be the value of $x$ ?

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Explanation

Choice A is correct. It’s given that the point $left parenthesis x comma 53 right parenthesis$ is a solution to the given system of inequalities in the xy-plane. This means that the coordinates of the point, when substituted for the variables $x$ and $y$ , make both of the inequalities in the system true. Substituting $53$ for $y$ in the inequality $y greater than 14$ yields $53 greater than 14$ , which is true. Substituting $53$ for $y$ in the inequality $4 x plus y less than 18$ yields $4 x plus 53 less than 18$ . Subtracting $53$ from both sides of this inequality yields $4 x less than negative 35$ . Dividing both sides of this inequality by $4$ yields $x less than negative 8.75$ . Therefore, $x$ must be a value less than $negative 8.75$ . Of the given choices, only $negative 9$ is less than $negative 8.75$ .

Choice B is incorrect. Substituting $negative 5$ for $x$ and $53$ for $y$ in the inequality $4 x plus y less than 18$ yields $4 (- 5) + 53 < 18$ , or $33 less than 18$ , which is not true.

Choice C is incorrect. Substituting $5$ for $x$ and $53$ for $y$ in the inequality $4 x plus y less than 18$ yields $4 left parenthesis 5 right parenthesis plus 53 less than 18$ , or $73 less than 18$ , which is not true.

Choice D is incorrect. Substituting $9$ for $x$ and $53$ for $y$ in the inequality $4 x plus y less than 18$ yields $4 left parenthesis 9 right parenthesis plus 53 less than 18$ , or $89 less than 18$ , which is not true.