Question

which ordered pairs are in the solution set of the system of linear inequalities?

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$$\begin{cases} y \\ge -\\frac{1}{2}x \\\\ y < \\frac{1}{2}x + 1 \\end{cases}$$

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• \\((5, -2), (3, -1), (4, -3)\\)
• \\((5, -2), (3, 1), (4, 2)\\)
• \\((5, -2), (-3, 1), (4, 2)\\)
• \\((5, -2), (3, 1), (-4, 2)\\)

Explanation:

🆕 New Concept Discovered: Solving Systems of Linear Inequalities
Finding points that satisfy multiple inequality conditions at once

Step 1: Understand the system of inequalities

To find which ordered pairs are in the solution set, each point \( (x, y) \) must satisfy both inequalities in the system:

1. \( y \geq -\frac{1}{2}x \)
2. \( y < \frac{1}{2}x + 1 \)

Alternatively, we can look at the graph. The solution set is the overlapping shaded region (the purple region where both the blue and red shaded areas meet).

• The boundary line \( y = -\frac{1}{2}x \) is solid, so points on this line can be included if they satisfy the second inequality.
• The boundary line \( y = \frac{1}{2}x + 1 \) is dashed, so points on this line are not included in the solution set.

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

All four options contain the point \( (5, -2) \). Let's verify it algebraically to be sure:

• Inequality 1:

$$ -2 \geq -\frac{1}{2}(5) $$

$$ -2 \geq -2.5 \quad \text{(True)} $$

• Inequality 2:

$$ -2 < \frac{1}{2}(5) + 1 $$

$$ -2 < 2.5 + 1 $$

$$ -2 < 3.5 \quad \text{(True)} $$

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

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Step 3: Test points from the options

Let's test the remaining points in the options to find the correct set.

Testing point \((3, 1)\):

• Inequality 1:

$$ 1 \geq -\frac{1}{2}(3) $$

$$ 1 \geq -1.5 \quad \text{(True)} $$

• Inequality 2:

$$ 1 < \frac{1}{2}(3) + 1 $$

$$ 1 < 1.5 + 1 $$

$$ 1 < 2.5 \quad \text{(True)} $$

Since both are true, \( (3, 1) \) is a solution. This eliminates the first option (which has \( (3, -1) \)) and the third option (which has \( (-3, 1) \)).

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Step 4: Test the remaining candidates \((4, 2)\) and \((-4, 2)\)

We now compare the second option containing \( (4, 2) \) and the fourth option containing \( (-4, 2) \).

Testing point \((4, 2)\):

• Inequality 1:

$$ 2 \geq -\frac{1}{2}(4) $$

$$ 2 \geq -2 \quad \text{(True)} $$

• Inequality 2:

$$ 2 < \frac{1}{2}(4) + 1 $$

$$ 2 < 2 + 1 $$

$$ 2 < 3 \quad \text{(True)} $$

Since both are true, \( (4, 2) \) is a solution.

Testing point \((-4, 2)\) for comparison:

• Inequality 1:

$$ 2 \geq -\frac{1}{2}(-4) $$

$$ 2 \geq 2 \quad \text{(True)} $$

• Inequality 2:

$$ 2 < \frac{1}{2}(-4) + 1 $$

$$ 2 < -2 + 1 $$

$$ 2 < -1 \quad \text{(False)} $$

Since \( (-4, 2) \) fails the second inequality, it is not a solution.

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Answer:

$$ (5, -2), (3, 1), (4, 2) $$