Question

which ordered pairs are in the solution set of the system of linear inequalities?
( y geq -\frac{1}{3}x + 2 )
( y < 2x + 3 )
( \bigcirc (2, 2), (3, 1), (4, 2) )
( \bigcirc (2, 2), (3, -1), (4, 1) )
( \bigcirc (2, 2), (1, -2), (0, 2) )
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Explanation:

Step1: Test (2,2) in both inequalities

First inequality: $2 \geq -\frac{1}{3}(2) + 2 = -\frac{2}{3} + 2 = \frac{4}{3}$ (True)
Second inequality: $2 < 2(2) + 3 = 4 + 3 = 7$ (True)

Step2: Test (3,1) in both inequalities

First inequality: $1 \geq -\frac{1}{3}(3) + 2 = -1 + 2 = 1$ (True, since $\geq$ includes equality)
Second inequality: $1 < 2(3) + 3 = 6 + 3 = 9$ (True)

Step3: Test (4,2) in both inequalities

First inequality: $2 \geq -\frac{1}{3}(4) + 2 = -\frac{4}{3} + 2 = \frac{2}{3}$ (True)
Second inequality: $2 < 2(4) + 3 = 8 + 3 = 11$ (True)

Step4: Test other pairs to eliminate options

Test (3,-1): $-1 \geq -\frac{1}{3}(3) + 2 = 1$ (False, so eliminate second option)
Test (1,-2): $-2 \geq -\frac{1}{3}(1) + 2 = \frac{5}{3}$ (False, so eliminate third option)

Answer:

(2, 2), (3, 1), (4, 2)