Systems with Parallel Boundary Lines
Identify systems with parallel boundary lines: when two inequalities have equal slopes but different intercepts, their boundaries never intersect and the feasible region has a special shape.
Key Concepts
Property When the boundary lines of a system are parallel (they have the exact same slope but different y intercepts), the shading dictates whether a solution exists: No Solution: If the shaded regions face away from each other and never overlap, there is no solution ($\emptyset$). Strip Region (Infinite Solutions): If the shaded regions face inward toward each other, the solution is the entire parallel "lane" trapped between the two boundary lines.
Examples Example 1 (No Solution): Solve the system $y x + 3$ and $y < x 1$. The first inequality shades everything ABOVE the higher parallel line. The second shades everything BELOW the lower parallel line. Because the regions point away from each other, they will never cross. There is No Solution. Example 2 (Strip Region): Solve the system $y \leq x + 5$ and $y \geq x + 1$. The first inequality shades BELOW the higher parallel line. The second shades ABOVE the lower parallel line. Because they shade inward toward each other, the solution is the endless parallel strip of space trapped between the two lines.
Explanation Parallel lines usually mean "no solution" when we are dealing with standard equations (because the lines never touch). But with inequalities, we are dealing with massive shaded areas! Imagine two parallel highways. If one rule says "shade everything North of Highway A", and the other says "shade everything South of Highway B", the shadings will never meet (No Solution). But if the rules shade the space in between the two highways, you end up with a massive, endless river of solutions flowing between them!
Common Questions
What happens to a system of inequalities when the boundary lines are parallel?
When boundary lines are parallel (same slope, different intercepts), they never intersect. The system may have no feasible region if the shaded areas are on opposite sides, or a strip-shaped feasible region if the shaded areas overlap between the two lines.
How do you identify parallel boundary lines in a system of inequalities?
Write each inequality's boundary in slope-intercept form y=mx+b and compare slopes. If all boundary lines share the same slope but different y-intercepts, they are parallel. Different slopes mean the lines intersect and the feasible region is bounded by those intersection points.
How does a system with parallel boundary lines differ from an inconsistent system of equations?
An inconsistent system of equations with parallel lines has no solution. A system of inequalities with parallel boundaries may still have a solution (the strip between the lines) depending on which sides are shaded. The inequalities allow a range of values rather than requiring exact equality.