Use the Systems of Inequalities graphed below to answer the questions.

Question

graph shows a dotted purple line going through (0,-2) and (4,0)
also graph shows a green line from (1.5,5) passing through (0,-2)

What is the inequality symbol of the green line?

What is the inequality symbol of the purple line?

Is the point (-3, -5) a solution to the system of inequalities?

Is the point (1, 6) a solution to the system of inequalities?

Answer 1

To determine the inequality symbols for the green and purple lines based on your description, we can analyze their characteristics and the regions they define:

1. Green Line: It passes through the points (1.5, 5) and (0, -2). Since this line appears to be solid (not a dotted line), it suggests that the inequality is "greater than or equal to." Therefore, the inequality symbol for the green line is: \[ y \geq mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.

2. Purple Line: It is described with a dotted line going through (0, -2) and (4, 0). The dotted line indicates that the region does not include the boundary itself, which typically means the inequality is "less than" or "greater than." Given how these lines interact (the green line being above the purple line in the visual you provided), we can conclude that this line represents "less than," meaning the inequality symbol for the purple line is: \[ y < mx + b \]

3. Point (-3, -5): To determine if this point is a solution to the system of inequalities, you will need to substitute it into both inequalities:

   • For the green line inequality (assume the equation is \( y \geq mx + b \)): Plugging \( x = -3 \) and \( y = -5 \): \[ -5 \geq m(-3) + b \]
   • For the purple line inequality (assume the equation is \( y < mx + b \)): Plugging \( x = -3 \) and \( y = -5 \): \[ -5 < m(-3) + b \] Without the exact equations, we cannot conclusively answer if (-3, -5) is a solution, but you can substitute \( x = -3 \) and \( y = -5 \) into the specific equations you have to check.

4. Point (1, 6): Similarly, substitute this point into both inequalities:

   • For the green line inequality: \[ 6 \geq m(1) + b \]
   • For the purple line inequality: \[ 6 < m(1) + b \] Again, evaluate these inequalities with the precise equations to determine if (1, 6) satisfies both conditions.

Overall, please check the actual equations corresponding to the lines to verify whether these points are solutions. Without the exact slopes and y-intercepts, we cannot definitively conclude the solutions.