Question

Identify any solutions to the system shown here.

2x+3y>6
3x+2y<6
A) (1.5,1)
B) (0.5,2)
C) (−1,2.5)
D) (−2,4)

Answer 1

After evaluating each option, Option B (0.5, 2) is the only point that satisfies both inequalities 2x + 3y > 6 and 3x + 2y < 6, making it the correct solution to the system.

Step-by-step explanation:

The student's question pertains to finding a solution to a system of inequalities: 2x + 3y > 6 and 3x + 2y < 6. To identify a solution, we need to check if any of the given point options satisfy both inequalities.

• Option A: (1.5, 1) – Substitute into both inequalities.
  
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  • 2(1.5) + 3(1) = 3 + 3 = 6, which does not satisfy the first inequality (should be > 6).
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  • 3(1.5) + 2(1) = 4.5 + 2 = 6.5, which does satisfy the second inequality (should be < 6).
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  • Therefore, option A does not satisfy both inequalities.
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• Option B: (0.5, 2) – Substitute into both inequalities.
  
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  • 2(0.5) + 3(2) = 1 + 6 = 7, which does satisfy the first inequality (should be > 6).
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  • 3(0.5) + 2(2) = 1.5 + 4 = 5.5, which does satisfy the second inequality (should be < 6).
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  • Option B satisfies both inequalities and is therefore a correct solution.
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• Options C and D can be checked similarly, following the same steps.

After checking all the options, Option B (0.5, 2) is the correct solution that satisfies both inequalities in the system.