Which point would be a solution to the system of linear inequalities shown below?

Question

y, is less than or equal to, one quarter, x, plus, 5, y, is greater than, 4, x, minus, 6
y≤
4
1
​
x+5y>4x−6
Answer
Multiple Choice Answers
left parenthesis, 8, comma, 5, right parenthesis(8,5)
left parenthesis, 12, comma, minus, 8, right parenthesis(12,−8)
left parenthesis, minus, 8, comma, minus, 10, right parenthesis(−8,−10)
left parenthesis, minus, 4, comma, 6, right parenthesis(−4,6)

Answer 1

To determine which of the given points is a solution to the system of inequalities

1. \( y \leq \frac{1}{4} x + 5 \)
2. \( y > 4x - 6 \)

we will evaluate each point by substituting the \( x \) and \( y \) values into the inequalities.

Checking each point:

1. Point (8, 5):

   • For \( y \leq \frac{1}{4} x + 5 \): \[ 5 \leq \frac{1}{4}(8) + 5 \implies 5 \leq 2 + 5 \implies 5 \leq 7 \quad \text{(True)} \]
   • For \( y > 4x - 6 \): \[ 5 > 4(8) - 6 \implies 5 > 32 - 6 \implies 5 > 26 \quad \text{(False)} \]

2. Point (12, -8):

   • For \( y \leq \frac{1}{4} x + 5 \): \[ -8 \leq \frac{1}{4}(12) + 5 \implies -8 \leq 3 + 5 \implies -8 \leq 8 \quad \text{(True)} \]
   • For \( y > 4x - 6 \): \[ -8 > 4(12) - 6 \implies -8 > 48 - 6 \implies -8 > 42 \quad \text{(False)} \]

3. Point (-8, -10):

   • For \( y \leq \frac{1}{4} x + 5 \): \[ -10 \leq \frac{1}{4}(-8) + 5 \implies -10 \leq -2 + 5 \implies -10 \leq 3 \quad \text{(True)} \]
   • For \( y > 4x - 6 \): \[ -10 > 4(-8) - 6 \implies -10 > -32 - 6 \implies -10 > -38 \quad \text{(True)} \]

4. Point (-4, 6):

   • For \( y \leq \frac{1}{4} x + 5 \): \[ 6 \leq \frac{1}{4}(-4) + 5 \implies 6 \leq -1 + 5 \implies 6 \leq 4 \quad \text{(False)} \]
   • For \( y > 4x - 6 \): \[ 6 > 4(-4) - 6 \implies 6 > -16 - 6 \implies 6 > -22 \quad \text{(True)} \]

Conclusion

The only point that satisfied both inequalities is Point (-8, -10).

Thus, the solution to the system of inequalities is (-8, -10).