1.4.2 Identification and Graphing of Feasible Regions¶

Determine and graph the region that satisfies all inequalities in a system simultaneously.

定义¶

A feasible region is the set of all points $(x, y)$ in the coordinate plane that simultaneously satisfy all inequalities in a system of linear inequalities. It represents the solution set of the system and is typically bounded by the boundary lines (or half-planes) defined by each inequality. The feasible region can be bounded (forming a closed polygon) or unbounded (extending infinitely in one or more directions). To identify a feasible region, one must: (1) graph each linear inequality as a boundary line (using solid lines for $\leq$ or $\geq$, and dashed lines for $<$ or $>$), (2) shade the appropriate half-plane for each inequality, and (3) find the intersection of all shaded regions. The vertices of the feasible region (corner points) are found at the intersections of the boundary lines and are critical for optimization problems.

核心公式¶

$ax + by \leq c$ (or $\geq, <, >$) represents a linear inequality in two variables
$y = mx + b$ is the boundary line of a linear inequality, where the line itself is included if the inequality is $\leq$ or $\geq$
$For a point $(x_0, y_0)$ to be in the feasible region: it must satisfy $a_1x_0 + b_1y_0 \leq c_1$ AND $a_2x_0 + b_2y_0 \leq c_2$ AND ... AND $a_nx_0 + b_ny_0 \leq c_n$$
$Intersection point of two boundary lines: solve the system $\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$ to find vertices of the feasible region$
$For optimization: the optimal value of an objective function $f(x,y) = px + qy$ occurs at a vertex of the feasible region (if the region is bounded)$

易错点¶

⚠️ Forgetting to use dashed lines for strict inequalities ($<$ or $>$) instead of solid lines, which incorrectly includes boundary points that should be excluded from the solution set
⚠️ Shading the wrong half-plane by not testing a point (such as the origin) to verify which side of the boundary line satisfies the inequality
⚠️ Failing to find all vertices of the feasible region by not solving the system of equations for all pairs of intersecting boundary lines, leading to incomplete identification of corner points needed for optimization
⚠️ Misinterpreting unbounded feasible regions as having no solution, when in fact an unbounded region is still a valid solution set that extends infinitely in certain directions