Answered

Which of the following points is a solution to the system of inequalities?



$\begin{array}{l}

y \leq -x + 1 \\

y > x

\end{array}$


A(-3, 5)
B(-2, 2)
C(-1, -3)
D(0, -1)

Verified Answer :

  • System of inequalities: \(y \leq -x + 1\) and \(y > x\).
  • Test each coordinate pair \((x, y)\) in the inequalities.
  • Point \((-3, 5)\) fails the first inequality (\(5 \leq 4\) is False).
  • Point \((-2, 2)\) satisfies both (\(2 \leq 3\) and \(2 > -2\)).
  • Point \((-1, -3)\) fails the second (\(-3 > -1\) is False).
  • Point \((0, -1)\) fails the second (\(-1 > 0\) is False).
  • Final Answer: \(\boxed{(-2, 2)}\).

Explanation

  1. Analyzing the System of Inequalities

We are analyzing the given system of two linear inequalities:
1) \(y \leq -x + 1\)
2) \(y > x\)

Our goal is to identify which of the following points \((-3, 5)\), \((-2, 2)\), \((-1, -3)\), and \((0, -1)\) satisfies both inequalities simultaneously. A point \((x, y)\) is a solution if it makes both statements true. We will test each point systematically by substituting the \(x\) and \(y\) coordinates into both inequalities.

  1. Testing Candidate Points
  • Testing point \((-3, 5)\):
  • Check inequality 1: \(5 \leq -(-3) + 1 = 3 + 1 = 4\). Since \(5 \leq 4\) is False, this point is not a solution.
  • Testing point \((-2, 2)\):
  • Check inequality 1: \(2 \leq -(-2) + 1 = 2 + 1 = 3\). Since \(2 \leq 3\) is True, this condition holds.
  • Check inequality 2: \(2 > -2\). Since \(2 > -2\) is True, this condition also holds.
  • Both conditions are satisfied, so \((-2, 2)\) is a solution.
  • Testing point \((-1, -3)\):
  • Check inequality 1: \(-3 \leq -(-1) + 1 = 1 + 1 = 2\). Since \(-3 \leq 2\) is True, this condition holds.
  • Check inequality 2: \(-3 > -1\). Since \(-3 > -1\) is False, this point is not a solution.
  • Testing point \((0, -1)\):
  • Check inequality 1: \(-1 \leq -(0) + 1 = 1\). Since \(-1 \leq 1\) is True, this condition holds.
  • Check inequality 2: \(-1 > 0\). Since \(-1 > 0\) is False, this point is not a solution.
  1. Conclusion

After evaluating all candidate points against the two inequalities, we found that only the point \((-2, 2)\) satisfies both requirements. Thus, the correct point is \((-2, 2)\). The final answer is \(\boxed{(-2, 2)}\).

Examples

This type of problem is essential in fields like linear programming, where you need to find an optimal point within a set of constraints. For example, if a business wants to maximize profit while restricted by resource capacity and market demand (represented by inequalities), they must find a value that satisfies all constraints simultaneously. Checking individual test cases helps verify which configurations meet the required operational boundaries.

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