Question

writing a description of how changing an inequality sign will change a solution
how will the solution of the system ( y > 2x + \frac{2}{3} ) and ( y < 2x + \frac{1}{3} ) change if the inequality sign on both inequalities is reversed to ( y < 2x + \frac{2}{3} ) and ( y > 2x + \frac{1}{3} )?

original system (( y > 2x + 2/3 ) and ( y < 2x + 1/3 )) has no solution - no ( y ) can be bigger than a larger value and smaller than a smaller one. when reversed to ( y < 2x + 2/3 ) and ( y > 2x + 1/3 ), the system now has infinitely many solutions: all ( (x, y) ) where ( y ) sits between the two parallel lines.

sample response: there is no solution to the system in its original form. there are no points in common. if the signs are reversed, the system has an intersection with an infinite number of solutions.
what did you include in your response? check all that apply.

• the shaded area would reverse on both inequalities.
• the graphs do not overlap until the signs are reversed.
• there are an infinite number of solutions in the system once the signs are reversed.
• there are no solutions before they are reversed.

Explanation:

• The original system requires $y$ to be greater than a larger value ($2x+\frac{2}{3}$) and smaller than a smaller value ($2x+\frac{1}{3}$), which is impossible, so there are no solutions.
• Reversing the signs means $y$ is less than the larger value and greater than the smaller value, creating an overlapping region between the two parallel lines with infinite solutions. Reversing the inequalities also reverses the shaded regions for each inequality.
• All listed statements align with these observations.

Answer:

• The shaded area would reverse on both inequalities.
• The graphs do not overlap until the signs are reversed.
• There are an infinite number of solutions in the system once the signs are reversed.
• There are no solutions before they are reversed.