A Cartesian coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. 2 dashed lines with arrows at both ends are plotted, and various regions of the graph are marked.

Question

Which point is a solution to the simultaneous inequalities y<12x
and y>13x+3
?

(1 point)
Responses

(25,12)
left parenthesis 25 comma 12 right parenthesis

(10,5)
left parenthesis 10 comma 5 right parenthesis

(3,6)
left parenthesis 3 comma 6 right parenthesis

(2,2)

Answer 1

To determine which point is a solution to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \), we can substitute the x and y values from each response into these inequalities.

1. For (25, 12):

   • Check \( y < 12x \): \[ 12 < 12 \times 25 \quad \text{(True, as } 12 < 300\text{)} \]
   • Check \( y > 13x + 3 \): \[ 12 > 13 \times 25 + 3 \quad \text{(False, as } 12 > 328\text{)} \]

2. For (10, 5):

   • Check \( y < 12x \): \[ 5 < 12 \times 10 \quad \text{(True, as } 5 < 120\text{)} \]
   • Check \( y > 13x + 3 \): \[ 5 > 13 \times 10 + 3 \quad \text{(False, as } 5 > 133\text{)} \]

3. For (3, 6):

   • Check \( y < 12x \): \[ 6 < 12 \times 3 \quad \text{(True, as } 6 < 36\text{)} \]
   • Check \( y > 13x + 3 \): \[ 6 > 13 \times 3 + 3 \quad \text{(False, as } 6 > 42\text{)} \]

4. For (2, 2):

   • Check \( y < 12x \): \[ 2 < 12 \times 2 \quad \text{(True, as } 2 < 24\text{)} \]
   • Check \( y > 13x + 3 \): \[ 2 > 13 \times 2 + 3 \quad \text{(False, as } 2 > 29\text{)} \]

None of the given points satisfy both inequalities simultaneously:

• (25, 12): Fails the second inequality.
• (10, 5): Fails the second inequality.
• (3, 6): Fails the second inequality.
• (2, 2): Fails the second inequality.

Therefore, none of the options provided are solutions to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \).