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A System of Linear Inequalities with No Solution
A system of linear inequalities can have no solution when the individual shaded regions for each inequality do not overlap anywhere on the coordinate plane. This situation can occur when the boundary lines of the inequalities are parallel and the inequalities require shading on opposite sides of their respective boundary lines. Because parallel lines never intersect and the shaded half-planes face away from each other, there is no region that satisfies both inequalities simultaneously. When graphing reveals no overlap between the shaded regions, the system has no solution.
This is analogous to an inconsistent system of linear equations, where parallel lines produce no common solution. However, unlike systems of equations — where parallel lines always mean no solution — parallel boundary lines in a system of inequalities do not always produce no solution. If both inequalities shade toward the region between the parallel lines, an overlapping strip can exist, and the system would have infinitely many solutions.
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