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How will the solution of the system y > 2x + Two-thirds and y < 2x + One-third change if the inequality sign on both inequalities is reversed to y < 2x + Two-thirds and
y > 2x + One-third?
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GPT-5 mini
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Original system: y > 2x + 2/3 and y < 2x + 1/3.
Since 2/3 > 1/3, these two inequalities would require
2x + 2/3 < y < 2x + 1/3,
which is impossible (equivalently 2/3 < 1/3). So the original system has no solution (empty set).
Reversed system: y < 2x + 2/3 and y > 2x + 1/3.
This requires
2x + 1/3 < y < 2x + 2/3,
so the solution is the (open) strip of all points between the two parallel lines y = 2x + 1/3 and y = 2x + 2/3 (boundaries excluded). For example, (x,y) = (0,0.5) satisfies the reversed system.
Since 2/3 > 1/3, these two inequalities would require
2x + 2/3 < y < 2x + 1/3,
which is impossible (equivalently 2/3 < 1/3). So the original system has no solution (empty set).
Reversed system: y < 2x + 2/3 and y > 2x + 1/3.
This requires
2x + 1/3 < y < 2x + 2/3,
so the solution is the (open) strip of all points between the two parallel lines y = 2x + 1/3 and y = 2x + 2/3 (boundaries excluded). For example, (x,y) = (0,0.5) satisfies the reversed system.
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