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Determine which of the following points are included in the solution set to the following system of linear inequalities. \[ \begin{aligned} x+3 y & >-8 \\ -2 x+3 y & \leq 1 \end{aligned} \] Answer \( (8,4) \) \( (0,0) \) \( (-8,-2) \) \( \left(-7,-\frac{1}{3}\right) \) \( (7,5) \)

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Jan 22,2025

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The points \( (8, 4) \), \( (0, 0) \), and \( (7, 5) \) are included in the solution set.

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To determine whether each point is included in the solution set, we need to test each point against the given inequalities. 1. For the point \( (8,4) \): - \( x + 3y = 8 + 3(4) = 8 + 12 = 20 > -8 \) (True) - \( -2x + 3y = -2(8) + 3(4) = -16 + 12 = -4 \leq 1 \) (True) This point satisfies both inequalities. 2. For the point \( (0,0) \): - \( x + 3y = 0 + 3(0) = 0 > -8 \) (True) - \( -2x + 3y = -2(0) + 3(0) = 0 \leq 1 \) (True) This point satisfies both inequalities. 3. For the point \( (-8,-2) \): - \( x + 3y = -8 + 3(-2) = -8 - 6 = -14 > -8 \) (False) It does not satisfy the first inequality. 4. For the point \( \left(-7,-\frac{1}{3}\right) \): - \( x + 3y = -7 + 3\left(-\frac{1}{3}\right) = -7 - 1 = -8 > -8 \) (False) It does not satisfy the first inequality. 5. For the point \( (7,5) \): - \( x + 3y = 7 + 3(5) = 7 + 15 = 22 > -8 \) (True) - \( -2x + 3y = -2(7) + 3(5) = -14 + 15 = 1 \leq 1 \) (True) This point satisfies both inequalities. Upon checking, the points included in the solution set are: \( (8,4), (0,0), \) and \( (7,5) \). The points \( (-8,-2) \) and \( \left(-7,-\frac{1}{3}\right) \) are not included.

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