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

Question

y, is greater than, minus, start fraction, 5, divided by, 2, end fraction, x, plus, 3, y, is greater than, 4, x, plus, 2
y>−
2
5
​
x+3y>4x+2
Answer
Multiple Choice Answers
left parenthesis, 12, comma, minus, 7, right parenthesis(12,−7)
left parenthesis, 2, comma, minus, 3, right parenthesis(2,−3)
left parenthesis, 2, comma, minus, 1, right parenthesis(2,−1)
left parenthesis, minus, 2, comma, 10, right parenthesis(−2,10)

Answer 1

To determine which points are solutions to the system of inequalities \( y > -\frac{2}{5}x + 3 \) and \( y > 4x + 2 \), we will evaluate each point against both inequalities.

1. Point \( (12, -7) \):

   • For \( y > -\frac{2}{5}x + 3 \): \[ -7 > -\frac{2}{5}(12) + 3 \Rightarrow -7 > -\frac{24}{5} + 3 \Rightarrow -7 > -\frac{24}{5} + \frac{15}{5} \Rightarrow -7 > -\frac{9}{5} \] This is false because \(-7 < -\frac{9}{5}\).

   • For \( y > 4x + 2 \): \[ -7 > 4(12) + 2 \Rightarrow -7 > 48 + 2 \Rightarrow -7 > 50 \] This is also false.

2. Point \( (2, -3) \):

   • For \( y > -\frac{2}{5}x + 3 \): \[ -3 > -\frac{2}{5}(2) + 3 \Rightarrow -3 > -\frac{4}{5} + 3 \Rightarrow -3 > -\frac{4}{5} + \frac{15}{5} \Rightarrow -3 > \frac{11}{5} \] This is false.

   • For \( y > 4x + 2 \): \[ -3 > 4(2) + 2 \Rightarrow -3 > 8 + 2 \Rightarrow -3 > 10 \] This is also false.

3. Point \( (2, -1) \):

   • For \( y > -\frac{2}{5}x + 3 \): \[ -1 > -\frac{2}{5}(2) + 3 \Rightarrow -1 > -\frac{4}{5} + 3 \Rightarrow -1 > -\frac{4}{5} + \frac{15}{5} \Rightarrow -1 > \frac{11}{5} \] This is false.

   • For \( y > 4x + 2 \): \[ -1 > 4(2) + 2 \Rightarrow -1 > 8 + 2 \Rightarrow -1 > 10 \] This is also false.

4. Point \( (-2, 10) \):

   • For \( y > -\frac{2}{5}x + 3 \): \[ 10 > -\frac{2}{5}(-2) + 3 \Rightarrow 10 > \frac{4}{5} + 3 \Rightarrow 10 > \frac{4}{5} + \frac{15}{5} \Rightarrow 10 > \frac{19}{5} \] This is true since \( 10 = \frac{50}{5} > \frac{19}{5} \).

   • For \( y > 4x + 2 \): \[ 10 > 4(-2) + 2 \Rightarrow 10 > -8 + 2 \Rightarrow 10 > -6 \] This is also true.

Therefore, the only point that satisfies both inequalities is \( (-2, 10) \).

The answer is: \( (-2, 10) \)