1.4.5 Solution and Verification of Systems of Inequalities

Solve systems of linear inequalities and verify that solutions satisfy all constraints.

定义

A system of linear inequalities is a set of two or more linear inequalities in one or more variables that must be satisfied simultaneously. A solution to a system of inequalities is a set of values for the variables that satisfies all inequalities in the system at the same time. For a system of inequalities in two variables, the solution set is typically represented as a region in the coordinate plane bounded by the boundary lines of each inequality. Verification involves substituting the proposed solution values back into each inequality to confirm that all inequalities are satisfied (the inequality statements are true).

核心公式

• \(\begin{cases} a_1x + b_1y \leq c_1 \ a_2x + b_2y \geq c_2 \ a_3x + b_3y < c_3 \end{cases}\)
• \(\text{Boundary line: } ax + by = c\)
• \(\text{Test point method: Substitute } (x_0, y_0) \text{ into each inequality to verify}\)
• \(\text{Solution region: Intersection of all half-planes satisfying each inequality}\)
• \(\text{Verification: If } (x_0, y_0) \text{ is a solution, then } a_1x_0 + b_1y_0 \leq c_1 \text{ AND } a_2x_0 + b_2y_0 \geq c_2 \text{ AND } a_3x_0 + b_3y_0 < c_3 \text{ (all true)}\)

易错点

• ⚠️ Forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number during algebraic manipulation of inequalities
• ⚠️ Incorrectly identifying the solution region by shading the wrong side of a boundary line, or confusing whether to use a solid line (≤ or ≥) versus a dashed line (< or >)
• ⚠️ Failing to verify that a proposed solution satisfies ALL inequalities in the system—checking only one or two inequalities instead of all of them
• ⚠️ Misinterpreting boundary lines as part of the solution when the inequality is strict (< or >) rather than inclusive (≤ or ≥)