N/A — Exam

Question 33 • Mathematics I

Which system of linear inequalities has the point 2 comma 1 in its solution set?

Answer

y is less than 1 half x y is less than or equal to negative 1 half x plus 2 Image with description On a coordinate plane, 2 straight lines are shown. The scale on both axes ranges from negative 5 to 5 in increments of 1. The first solid line has a negative slope and goes through (0 comma 2) and (4 comma 0). Everything below the line is shaded. The second dashed line has a positive slope and goes through (negative 4 comma negative 2) and (0 comma 0). Everything below the line is shaded.

y is less than negative 1 half x plus 3 y is less than 1 half x Image with description On a coordinate plane, 2 lines are shown. The scale on both axes ranges from negative 5 to 5 in increments of 1. The first solid straight line has a negative slope and goes through (0 comma 3) and (4 comma 1). Everything below the line is shaded. The second dashed straight line has a positive slope and goes through (0 comma 0) and (2 comma 1). Everything below and to the right of the line is shaded.

y is less than negative x plus 3 y is less than or equal to 1 half x plus 3 Image with description On a coordinate plane, 2 lines are shown. The scale on both axes ranges from negative 5 to 5. The first solid straight line has a positive slope and goes through (negative 4 comma 1) and (0 comma 3). Everything below the line is shaded. The second dashed straight line has a negative slope and goes through (0 comma 3) and (3 comma 0). Everything to the left of the solid line is shaded.

y is less than or equal to negative x plus 3 y is less than or equal to 1 half x plus 3 Image with description On a coordinate plane, 2 solid straight lines are shown. The scale on both axes ranges from negative 5 to 5 in increments of 1. The first solid line has a positive slope and goes through (negative 4 comma 1) and (0 comma 3). Everything below the line is shaded. The second solid line has a negative slope and goes through (0 comma 3) and (3 comma 0). Everything below and to the left of the line is shaded.

Explanation

Substitute x = 2 and y = 1 into choice D. The first inequality becomes 1 ≤ −2 + 3, or 1 ≤ 1, which is true; the second becomes 1 ≤ ½(2) + 3, or 1 ≤ 4, which is also true. Therefore, choice D, y ≤ −x + 3 and y ≤ ½x + 3, includes the point (2, 1).

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