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

In a set of four consecutive odd integers, where the integers are ordered from least to greatest, the first integer is represented by $x$ . The product of $12$ and the fourth odd integer is at most $26$ less than the sum of the first and third odd integers. Which inequality represents this situation?

$12 (x + 6) \leq x + (x + 4) - 26$

$12 (x + 6) \geq 26 - (x + (x + 4))$

$12 (x + 4) \leq x + (x + 3) - 26$

$12 (x + 4) \geq 26 - (x + (x + 3))$

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Explanation

Choice A is correct. It’s given that the four odd integers are consecutive, ordered from least to greatest, and that the first odd integer is represented by $x$ . It follows that the second odd integer is represented by $x plus 2$ , the third odd integer is represented by $x plus 4$ , and the fourth odd integer is represented by $x plus 6$ . Therefore, the product of $12$ and the fourth odd integer is represented by $12 left parenthesis x plus 6 right parenthesis$ , and $26$ less than the sum of the first and third odd integers is represented by $x + (x + 4) - 26$ . Since the product of $12$ and the fourth odd integer is at most $26$ less than the sum of the first and third odd integers, it follows that $12 (x + 6) \leq x + (x + 4) - 26$ .

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

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

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