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1.4.1 Representation and Conversion of Inequalities in Systems

Express and convert multiple linear inequalities in algebraic and graphical forms to represent 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. Each inequality represents a constraint, and the solution set is the intersection of all individual solution regions. In two variables, a linear inequality of the form \(ax + by \leq c\) (or \(<, >, \geq\)) represents a half-plane in the coordinate plane, bounded by the line \(ax + by = c\). The solution to a system is the region where all inequalities are true at the same time. This region can be represented algebraically as a set of ordered pairs satisfying all conditions, or graphically as the overlapping area of all half-planes. Systems of linear inequalities are fundamental in linear programming and optimization problems, where we seek to maximize or minimize an objective function subject to given constraints.

核心公式

  • \(ax + by \leq c\) (or \(<, >, \geq\))
  • \(\begin{cases} a_1x + b_1y \leq c_1 \\ a_2x + b_2y \leq c_2 \\ \vdots \\ a_nx + b_ny \leq c_n \end{cases}\)
  • \(\text{Solution Set} = \{(x,y) \in \mathbb{R}^2 : a_1x + b_1y \leq c_1 \text{ AND } a_2x + b_2y \leq c_2 \text{ AND } \cdots \}\)
  • \(y \leq mx + b\) (or \(y \geq mx + b\)) for slope-intercept form
  • \(\text{Feasible Region} = \bigcap_{i=1}^{n} \{(x,y) : a_ix + b_iy \leq c_i\}\)

易错点

  • ⚠️ Incorrectly flipping the inequality sign when multiplying or dividing by a negative number, especially when converting inequalities to standard form or solving for a variable
  • ⚠️ Confusing the boundary line representation: using a solid line for strict inequalities (\(<\) or \(>\)) instead of dashed lines, or vice versa, leading to incorrect graphical representations
  • ⚠️ Shading the wrong region on the graph by not testing a point to verify which side of the boundary line satisfies the inequality
  • ⚠️ Forgetting that the solution to a system requires ALL inequalities to be satisfied simultaneously (intersection), not just one (union), leading to incorrect identification of the feasible region