Question

Determine which of the graphs is the graph of a system of inequalities with no solutions.

a. The solid line passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis
O comma 2 right parenthesis. The area above the line is shaded solid blue. The area below the line is marked with a diagonal line pattern. The area above the line is labeled y is greater than or equal to x plus
2. The area below the line is labeled y is less than or equal to x plus 2.

b. The first line is labeled y is greater than or equal to 2 x minus 1. This line passes through left parenthesis negative 1 comma negative 3 right parenthesis and left parenthesis 2 comma 4 right parenthesis. The area to the left of this line is shaded. The other line is labeled y is less than or equal to 3 x minus 2. This line passes through left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 2 comma 3 right parenthesis. The area to the right of this line is marked with a diagonal line pattern. As the lines cross at left parenthesis 1 comma 1 right parenthesis, the area in quadrant 1 between the lines is both shaded and marked with a diagonal line pattern. The area between the lines below the intersection is unmarked.

c. The solid line passes through coordinates left parenthesis negative 6 comma 1 right parenthesis and left parenthesis 18 comma 9 right parenthesis. The dashed line passes through graphed point left parenthesis 10 comma 5 right parenthesis and coordinates left parenthesis 18 comma 9 right parenthesis. The area above the solid line but also above the dashed line is shaded. The area below the dashed line but also below the solid line is filled with a dot pattern. The area above the solid line but also below the dashed line is filled with a triangle pattern. The first graphed point has coordinates left parenthesis 2 comma 2 right parenthesis. The second graphed point has coordinates left parenthesis 3 comma 6 right parenthesis. The third graphed point has coordinates left parenthesis 10 comma 5 right parenthesis. The fourth graphed point has coordinates left parenthesis 25 comma 12 right parenthesis.

d. The first solid line is labeled y is greater than or equal to 2 x plus 1. This line passes through left parenthesis negative 2 comma negative 3 right parenthesis and left parenthesis 1 comma 3 right parenthesis. The area to the left of this line is marked with a diagonal line pattern. The other solid line is labeled as y is greater than or equal to 2 x minus 1. This line passes through left parenthesis negative 1 comma negative 3 right parenthesis and left parenthesis 2 comma 3 right parenthesis. The area to the left of this line is shaded. Therefore the area to the left of the first line is both shaded and marked with a diagonal line pattern. The area between the lines is only shaded.

Answer 1

d

Reason: the two lines are parallel (same slope 2) with different intercepts. If the system requires y ≥ 2x + 1 and y ≤ 2x − 1, then 2x + 1 ≤ y ≤ 2x − 1 is impossible (2x + 1 ≤ 2x − 1 would imply 1 ≤ −1). Thus there is no point satisfying both inequalities.