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

$y is greater than 2 x minus 1, and, 2 x is greater than 5$

Which of the following consists of the y-coordinates of all the points that satisfy the system of inequalities above?

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Explanation

Choice B is correct. Subtracting the same number from each side of an inequality gives an equivalent inequality. Hence, subtracting 1 from each side of the inequality $2 x is greater than 5$ gives $2 x minus 1, is greater than 4$ . So the given system of inequalities is equivalent to the system of inequalities $y is greater than, 2 x minus 1$ and $2 x minus 1, is greater than 4$ , which can be rewritten as $y is greater than, 2 x minus 1, which is greater than 4$ . Using the transitive property of inequalities, it follows that $y is greater than 4$ .

Choice A is incorrect because there are points with a y-coordinate less than 6 that satisfy the given system of inequalities. For example, $the point with coordinates 3 comma 5 point 5$ satisfies both inequalities. Choice C is incorrect. This may result from solving the inequality $2 x is greater than 5$ for x, then replacing x with y. Choice D is incorrect because this inequality allows y-values that are not the y-coordinate of any point that satisfies both inequalities. For example, $y equals 2$ is contained in the set $y is greater than three halves$ ; however, if 2 is substituted into the first inequality for y, the result is $x is less than three halves$ . This cannot be true because the second inequality gives $x is greater than five halves$ .