Question

1. for a system of inequalities to have no solution, which statement must be true?

all systems of inequalities have solutions.
the boundary lines must have different slopes.
the boundary lines must have the same slope.
the shaded regions must overlap.

Explanation:

• Analyze Option 1: "All systems of inequalities have solutions" is false, as some systems (e.g., conflicting inequalities) have no solution.
• Analyze Option 2: Different slopes mean lines intersect, so the system would have a solution (the intersection region), so this is incorrect.
• Analyze Option 3: For a system of inequalities to have no solution, the boundary lines often are parallel (same slope) and the shaded regions don’t overlap. If slopes differ, lines intersect, so there’s a solution region. Thus, same slope is a necessary condition for no solution (when regions don’t overlap).
• Analyze Option 4: If shaded regions overlap, the system has solutions (the overlap is the solution set), so this is incorrect.

Answer:

C. The boundary lines must have the same slope.