Question

select the correct answer.

which point is a solution to the given system of inequalities?
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$$\begin{aligned} x + 4y &> 12 \\\\ 3y &> x + 6 \\end{aligned}$$

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• (-2,3)
• (-4,6)
• (5,2)
• (0,3)

Explanation:

🆕 New Concept Discovered: Systems of Linear Inequalities
A point must make all inequalities true.

Step 1: Understand the conditions

To find which point is a solution to the system, we must substitute the coordinates \( (x, y) \) of each option into both inequalities. A point is a solution only if it makes both statements true.

The system of inequalities is:

1. \( x + 4y > 12 \)
2. \( 3y > x + 6 \)

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

Substitute \( x = -2 \) and \( y = 3 \):

• First inequality:

$$ -2 + 4(3) > 12 $$

$$ -2 + 12 > 12 $$

$$ 10 > 12 \quad \text{(False)} $$

Since the first inequality is false, \( (-2, 3) \) is not a solution.

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Step 3: Test the point \( (-4, 6) \)

Substitute \( x = -4 \) and \( y = 6 \):

• First inequality:

$$ -4 + 4(6) > 12 $$

$$ -4 + 24 > 12 $$

$$ 20 > 12 \quad \text{(True)} $$

• Second inequality:

$$ 3(6) > -4 + 6 $$

$$ 18 > 2 \quad \text{(True)} $$

Since both inequalities are true, \( (-4, 6) \) is a solution.

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Step 4: Verify remaining points for completeness

• Test \( (5, 2) \):

$$ 5 + 4(2) > 12 \implies 13 > 12 \quad \text{(True)} $$

$$ 3(2) > 5 + 6 \implies 6 > 11 \quad \text{(False)} $$

• Test \( (0, 3) \):

$$ 0 + 4(3) > 12 \implies 12 > 12 \quad \text{(False, because 12 is not strictly greater than 12)} $$

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Answer:

\( (-4,6) \)