Question

the graph represents a system of inequalities for the scenario where x is the number of confirmed parties on a tour and y is the number of required copies of the itinerary. which constraint could be part of this system of equations?

• at least 2 copies are required per tour.
• more than 2 copies are required per tour.
• the number of confirmed parties must be at least double the number of required copies.
• the number of confirmed parties must be no more than double the number of required copies.

Explanation:

Step1: Find line equation

The blue line passes through (0,2) and (-2,1). Slope $m=\frac{2-1}{0-(-2)}=\frac{1}{2}$. Equation: $y=\frac{1}{2}x+2$, rearranged to $x-2y\geq-4$ (not needed for options). Now check each option against the shaded region.

Step2: Analyze option 1

"At least 2 copies are required per tour" means $y\geq2$. But the shaded region includes points where $y<2$ (e.g., (0,1)), so this is invalid.

Step3: Analyze option 2

"More than 2 copies are required per tour" means $y>2$. The shaded region includes points where $y<2$ (e.g., (0,1)), so this is invalid.

Step4: Analyze option 3

"The number of confirmed parties must be at least double the number of required copies" means $x\geq2y$, or $x-2y\geq0$. Test point (0,2): $0-4=-4<0$, but (0,2) is in the shaded region, so this is invalid.

Step5: Analyze option 4

"The number of confirmed parties must be no more than double the number of required copies" means $x\leq2y$, or $x-2y\leq0$. Test point (0,2): $0-4=-4\leq0$ (valid, in shaded region). Test point (2,2): $2-4=-2\leq0$ (valid, in shaded region). Test point (-2,1): $-2-2=-4\leq0$ (valid, in shaded region). This matches the shaded region.

Answer:

The number of confirmed parties must be no more than double the number of required copies.