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

$2 l + 2 w \leq 27$ $2 l + 2 w \leq 27$

A rectangle has length $l$ $l$ and width $w$ $w$ . The inequality gives the possible values of $l$ $l$ and $w$ $w$ for which the perimeter of this rectangle is less than or equal to $27$ $27$ . Which statement is the best interpretation of $(l, w) = (8, 3)$ $left parenthesis script l comma w right parenthesis equals left parenthesis 8 comma 3 right parenthesis$ in this context?

If the rectangle has length $3$ and width $8$ , its perimeter is less than or equal to $27$ .

If the rectangle has length $8$ and width $3$ , its perimeter is less than or equal to $27$ .

If the rectangle has length $3$ and width $8$ , its perimeter is greater than or equal to $27$ .

If the rectangle has length $8$ and width $3$ , its perimeter is greater than or equal to $27$ .

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Explanation

Choice B is correct. It’s given that a rectangle has length $l$ $l$ and width $w$ $w$ , and the inequality $2 l + 2 w \leq 27$ $2 l + 2 w \leq 27$ gives the possible values of $l$ $l$ and $w$ $w$ for which the perimeter of this rectangle is less than or equal to $27$ $27$ . To determine the best interpretation of $(l, w) = (8, 3)$ $left parenthesis script l comma w right parenthesis equals left parenthesis 8 comma 3 right parenthesis$ in this context, the values can be substituted in the given inequality. Substituting $8$ $8$ for $l$ $l$ and $3$ $3$ for $w$ $w$ in this inequality yields $2 (8) + 2 (3) \leq 27$ $2 (8) + 2 (3) \leq 27$ , which is equivalent to $16 + 6 \leq 27$ $16 + 6 \leq 27$ , or $22 \leq 27$ $22 less than or equals 27$ . Since this inequality is true, if the rectangle has length $8$ $8$ and width $3$ $3$ , its perimeter is less than or equal to $27$ $27$ .

Choice A is incorrect. The interpretation of $(l, w) = (8, 3)$ $left parenthesis script l comma w right parenthesis equals left parenthesis 8 comma 3 right parenthesis$ implies that the rectangle has length $8$ $8$ and width $3$ $3$ , not length $3$ $3$ and width $8$ $8$ .

Choice C is incorrect. The interpretation of $(l, w) = (8, 3)$ $left parenthesis script l comma w right parenthesis equals left parenthesis 8 comma 3 right parenthesis$ implies that the rectangle has length $8$ $8$ and width $3$ $3$ , not length $3$ $3$ and width $8$ $8$ .

Choice D is incorrect and may result from conceptual or calculation errors.