1. Which of the following inequalities is graphed in the xy-plane above?
Answer: C
y ≥ -1/2x + 2
The inequality graphed in the xy-plane indicates that the region above the line is included, which corresponds to the option y ≥ -1/2x + 2.
A) y ≥ 1/2x + 2
This option represents a line with a slope of 1/2, which would rise rather than fall. The shaded region would be above this line, contradicting the downward slope observed in the graph. Thus, this option is incorrect.
B) y ≤ 1/2x + 2
Similar to option A, this choice has a slope of 1/2, leading to a shaded region below the line. The graph instead shows shading above a line with a negative slope, making this option incorrect.
C) y ≥ -1/2x + 2
This option accurately represents the inequality shown in the graph. The line has a slope of -1/2, indicating a downward trend, and the shading above the line confirms that values of y are greater than or equal to the expression, thus making this the correct answer.
D) y ≤ -1/2x + 2
This option describes a line with a slope of -1/2 as well, but suggests that the region below the line is shaded. The graph displays the opposite, with the area above the line shaded. Therefore, this option is incorrect.
Conclusion
The correct answer, y ≥ -1/2x + 2, matches the characteristics of the graph with its downward slope and shading above the line. In contrast, the other options either represent the incorrect slope or shading direction, confirming that they do not correspond to the graph provided.