1.2.3 Graphical Representation of Inequalities¶

Represent linear inequality solutions on number lines and coordinate planes using open/closed intervals and shading.

定义¶

Graphical representation of inequalities is the visual depiction of solution sets for linear inequalities on both number lines and coordinate planes. On a number line, solutions are represented using open circles (for strict inequalities $<$ or $>$) or closed circles (for non-strict inequalities $\leq$ or $\geq$), with shading or arrows indicating the direction of the solution set. On a coordinate plane, linear inequalities in two variables are represented by shading entire regions bounded by a boundary line. The boundary line itself is drawn as a dashed line for strict inequalities (where the boundary is not included) or a solid line for non-strict inequalities (where the boundary is included). The shaded region represents all ordered pairs $(x, y)$ that satisfy the inequality.

核心公式¶

$ax + b < c$ or $ax + b > c$ (strict inequalities use dashed boundary lines)
$ax + b \leq c$ or $ax + b \geq c$ (non-strict inequalities use solid boundary lines)
$ax + by < c$ or $ax + by > c$ (two-variable strict inequalities)
$ax + by \leq c$ or $ax + by \geq c$ (two-variable non-strict inequalities)
$Interval notation: $[a, b]$ (closed), $(a, b)$ (open), $[a, b)$ or $(a, b]$ (half-open)$

易错点¶

⚠️ ["Using a solid line instead of a dashed line (or vice versa) when graphing inequalities—students often forget that strict inequalities ($<$, $>$) require dashed boundary lines while non-strict inequalities ($\leq$, $\geq$) require solid lines", "Shading the wrong region on a coordinate plane—students may shade the region that does NOT satisfy the inequality, often because they fail to test a point from each region or misinterpret the inequality symbol", "Incorrectly representing endpoints on number lines—confusing when to use open circles versus closed circles, particularly when dealing with compound inequalities or when the endpoint is not included in the solution set", "Forgetting to reverse the inequality sign when multiplying or dividing by a negative number, leading to incorrect shading direction on both number lines and coordinate planes"]