Which of the following inequalities is graphed in the xy-plane above?
−x≤y≤3x
The inequality −x ≤ y ≤ 3x indicates that y is bounded above by the line y = 3x and below by the line y = -x. This means that for any given x-value, y can take on values between these two lines, which shapes the region represented in the xy-plane.
This choice states that y is bounded below by the line y = x and above by the line y = 3x. The graph of this inequality would depict a region that starts from the line y = x, which is incorrect since the provided graph has y starting from y = -x instead. Thus, this choice does not accurately represent the graphed region.
This option suggests that y is bounded above by x and below by 3x. However, this interpretation implies that y is less than or equal to x, which is not consistent with the graph showing a region above the line y = -x. Hence, this choice is also incorrect.
This is the correct choice, as it accurately captures the region between the lines y = -x and y = 3x. The graph illustrates that for any value of x, y can indeed take values between these two lines, matching the conditions set by the inequality.
This choice indicates that y is bounded above by -x and below by 3x. This scenario does not match the provided graph, as it would represent a region below the line y = -x rather than above it. Therefore, this option is incorrect.
The inequality −x ≤ y ≤ 3x correctly describes the region between the lines y = -x and y = 3x in the xy-plane, allowing y to vary within these bounds. The other options either misrepresent the relationships between x and y or depict regions that do not align with the graph provided, confirming that choice C is the only accurate representation.
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