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If the system of inequalities \(y \geq 2x + 1\) and \(y > \frac{1}{2}x - 1\) ...


Question

If the system of inequalities \(y \geq 2x + 1\) and \(y > \frac{1}{2}x - 1\) is graphed in the xy-plane above, which quadrant contains no solutions to the system? 

Options

A)
Quadrant II
B)
Quadrant III
C)
Quadrant IV
D)
There are solutions in all four quadrants.

The correct answer is C.

Explanation:

Choice C is correct. To determine which quadrant does not contain any solutions to the system of inequalities, graph the inequalities. Graph the inequality y \(\geq\) 2x + 1 by drawing a line through the y-intercept (0, 1) and the point (1, 3), as shown. The solutions to this inequality are all points contained on and above this line. Graph the inequality y > \(\frac{1}{2}\)x - 1 by drawing a dashed line through the y-intercept (0, -1) and the point (2, 0), as shown. The solutions to this inequality are all points above this dashed line.

The solution to the system of inequalities is the intersection of the regions above the graphs of both lines. It can be seen that the solutions only include points in quadrants I, II, and III, and do not include any points in quadrant IV. Choices A and B are incorrect because quadrants II and III contain solutions to the system of inequalities, as shown in the figure above. Choice D is incorrect because there are no solutions in quadrant IV.


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