Question

On a coordinate plane, 2 straight lines are shown. The first solid line has a positive slope and goes through (0, 3) and (3, 4). Everything above the line is shaded. The second dashed line has a positive slope and goes through (0, negative 2) and (1, 1). Everything to the right of the line is shaded.
Which system of linear inequalities is represented by the graph?

y > One-thirdx + 3 and 3x – y > 2
y > One-halfx + 3 and 3x – y > 2
y > One-thirdx + 3 and 3x + y > 2
y > One-thirdx + 3 and 2x – y > 2

Answer 1

First line through (0,3) and (3,4) has slope (4−3)/(3−0)=1/3, so its equation is y = (1/3)x + 3. It's solid and shaded above → y ≥ (1/3)x + 3 (given choices use >).

Second line through (0,−2) and (1,1) has slope (1−(−2))/(1−0)=3, so y = 3x − 2. "Right of" a positively sloped line is the region below it, i.e. y < 3x − 2, which rearranges to 3x − y > 2 (strict because the line is dashed).

Thus the system is: y > (1/3)x + 3 and 3x − y > 2.