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Determine which of the following points are included in the solution set to the following linear inequality. \[ \begin{array}{l}-5 x+4 y \geq-5\end{array} \] Answer \( \square(4,-5) \) \( \square\left(4, \frac{15}{4}\right) \) \( \square(0,0) \) \( \square(-4,4) \) \( \square(7,2) \)

Ask by Luna Lang. in the United States
Jan 22,2025

Real Tutor Solution

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Answer

The points that satisfy the inequality \(-5x + 4y \geq -5\) are: - \(\left(4, \frac{15}{4}\right)\) - \((0, 0)\) - \((-4, 4)\)

Solution

To determine which points satisfy the inequality: \[ -5x + 4y \geq -5 \] we will substitute each point \((x, y)\) into the inequality and check if the condition holds. 1. **Point \((4, -5)\):** \[ -5(4) + 4(-5) = -20 - 20 = -40 \quad \text{(Not } \geq -5\text{)} \] **Does not satisfy the inequality.** 2. **Point \(\left(4, \frac{15}{4}\right)\):** \[ -5(4) + 4\left(\frac{15}{4}\right) = -20 + 15 = -5 \quad \text{(Equal to } -5\text{)} \] **Satisfies the inequality.** 3. **Point \((0, 0)\):** \[ -5(0) + 4(0) = 0 \quad \text{(Greater than } -5\text{)} \] **Satisfies the inequality.** 4. **Point \((-4, 4)\):** \[ -5(-4) + 4(4) = 20 + 16 = 36 \quad \text{(Greater than } -5\text{)} \] **Satisfies the inequality.** 5. **Point \((7, 2)\):** \[ -5(7) + 4(2) = -35 + 8 = -27 \quad \text{(Not } \geq -5\text{)} \] **Does not satisfy the inequality.** **Conclusion:** The points that **are included** in the solution set to the inequality \(-5x + 4y \geq -5\) are: - \(\left(4, \frac{15}{4}\right)\) - \((0, 0)\) - \((-4, 4)\)

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The Deep Dive

To determine whether the points are included in the solution set of the inequality \(-5x + 4y \geq -5\), we substitute each point into the inequality and check if it holds true. 1. For the point \((4, -5)\): \[-5(4) + 4(-5) = -20 - 20 = -40 \quad (Not \geq -5)\] So, this point is not included. 2. For the point \(\left(4, \frac{15}{4}\right)\): \[-5(4) + 4\left(\frac{15}{4}\right) = -20 + 15 = -5 \quad (is \geq -5)\] So, this point is included. 3. For the point \((0, 0)\): \[-5(0) + 4(0) = 0 \quad (is \geq -5)\] So, this point is included. 4. For the point \((-4, 4)\): \[-5(-4) + 4(4) = 20 + 16 = 36 \quad (is \geq -5)\] So, this point is included. 5. For the point \((7, 2)\): \[-5(7) + 4(2) = -35 + 8 = -27 \quad (Not \geq -5)\] So, this point is not included. In summary: - \((4, -5)\): Not included - \((4, \frac{15}{4})\): Included - \((0, 0)\): Included - \((-4, 4)\): Included - \((7, 2)\): Not included The points included in the solution set are \(\left(4, \frac{15}{4}\right)\), \((0, 0)\), and \((-4, 4)\).

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