Question

select the correct answer.

which point is a solution to this system of inequalities?

\\(y \le \frac{1}{2}x - 3\\)
\\(y + 2x > 6\\)

(7, -8)
(5, -2)
(2, -3)
(4, 1)

Explanation:

🆕 New Concept Discovered: Solving Systems of Linear Inequalities
Testing points in multiple inequality boundaries

Step 1: Understand the conditions

To find which point is a solution to the system of inequalities, we must test the given coordinate points \( (x, y) \) in both inequalities. A point is a solution only if it makes both statements true.

The system is:

1. \( y \le \frac{1}{2}x - 3 \)
2. \( y + 2x > 6 \)

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Step 2: Test the point \( (7, -8) \)

Substitute \( x = 7 \) and \( y = -8 \):

• First inequality:

$$ -8 \le \frac{1}{2}(7) - 3 $$

$$ -8 \le 3.5 - 3 $$

$$ -8 \le 0.5 \quad \text{(True)} $$

• Second inequality:

$$ -8 + 2(7) > 6 $$

$$ -8 + 14 > 6 $$

$$ 6 > 6 \quad \text{(False, because 6 is equal to 6, not strictly greater than)} $$

Since the second inequality is false, \( (7, -8) \) is not a solution.

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Step 3: Test the point \( (5, -2) \)

Substitute \( x = 5 \) and \( y = -2 \):

• First inequality:

$$ -2 \le \frac{1}{2}(5) - 3 $$

$$ -2 \le 2.5 - 3 $$

$$ -2 \le -0.5 \quad \text{(True)} $$

• Second inequality:

$$ -2 + 2(5) > 6 $$

$$ -2 + 10 > 6 $$

$$ 8 > 6 \quad \text{(True)} $$

Since both inequalities are true, \( (5, -2) \) is a solution.

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Step 4: Verify remaining options for certainty

• Test \( (2, -3) \):
• First inequality: \( -3 \le \frac{1}{2}(2) - 3 \implies -3 \le -2 \) (True)
• Second inequality: \( -3 + 2(2) > 6 \implies 1 > 6 \) (False)

• Test \( (4, 1) \):
• First inequality: \( 1 \le \frac{1}{2}(4) - 3 \implies 1 \le -1 \) (False)

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Answer:

\( (5, -2) \)