Use the image to answer the question.

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.

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

(1 point)
Responses

(2,2)
left parenthesis 2 comma 2 right parenthesis

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

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

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

To determine which point is a solution to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \), we need to evaluate each point against both inequalities.

  1. Point (2, 2):

    • For \( y < 12x \): \[ 2 < 12(2) \Rightarrow 2 < 24 \quad \text{(True)} \]
    • For \( y > 13x + 3 \): \[ 2 > 13(2) + 3 \Rightarrow 2 > 26 + 3 \Rightarrow 2 > 29 \quad \text{(False)} \]
  2. Point (3, 6):

    • For \( y < 12x \): \[ 6 < 12(3) \Rightarrow 6 < 36 \quad \text{(True)} \]
    • For \( y > 13x + 3 \): \[ 6 > 13(3) + 3 \Rightarrow 6 > 39 + 3 \Rightarrow 6 > 42 \quad \text{(False)} \]
  3. Point (25, 12):

    • For \( y < 12x \): \[ 12 < 12(25) \Rightarrow 12 < 300 \quad \text{(True)} \]
    • For \( y > 13x + 3 \): \[ 12 > 13(25) + 3 \Rightarrow 12 > 325 + 3 \Rightarrow 12 > 328 \quad \text{(False)} \]
  4. Point (10, 5):

    • For \( y < 12x \): \[ 5 < 12(10) \Rightarrow 5 < 120 \quad \text{(True)} \]
    • For \( y > 13x + 3 \): \[ 5 > 13(10) + 3 \Rightarrow 5 > 130 + 3 \Rightarrow 5 > 133 \quad \text{(False)} \]

None of the points satisfy both inequalities \( y < 12x \) and \( y > 13x + 3 \). Therefore, it appears that none of the provided options are solutions to the simultaneous inequalities.

If the graph image is available, please make sure to confirm the arrangement of the lines, as the above analysis is based solely on the algebraic interpretation of the inequalities given.