Linear Inequalities - Inequalities

Linear Inequalities are mathematical expressions involving variables where the relationship is expressed using inequality symbols like $<$, $>$, $\le$, or $\ge$. A linear inequality in one variable $x$ typically takes the form $ax + b < 0$, while in two variables $x$ and $y$ it looks like $ax + by + c > 0$.

Algebraic Rules: (1) Equal numbers can be added to or subtracted from both sides without changing the inequality sign. (2) Multiplying or dividing both sides by a positive number preserves the inequality sign. (3) Multiplying or dividing both sides by a negative number reverses the direction of the inequality sign ($<$ becomes $>$, and vice versa).

Interval Notation: Solutions are often written as intervals. A closed interval $[a, b]$ includes the endpoints ($a \le x \le b$), whereas an open interval $(a, b)$ excludes them ($a < x < b$). Semi-open/closed intervals like $[a, b)$ include one endpoint and exclude the other.

Representation on a Number Line: For inequalities in one variable, the solution set is graphed as a ray or segment. A 'hollow/open circle' $\circ$ is used to denote that the boundary value is excluded (strict inequality like $<$ or $>$). A 'darkened/solid circle' $\bullet$ is used to denote that the boundary value is included (slack inequality like $\le$ or $\ge$).

Graphical Solution in Two Variables: A linear inequality $ax + by < c$ divides the Cartesian plane into two half-planes. To find the solution region, first graph the boundary line $ax + by = c$. If the inequality is strict ($<$ or $>$), draw a dashed or broken line; if it is slack ($\le$ or $\ge$), draw a solid line. Visually, the solution set is represented by shading the appropriate half-plane.

Trial Point Method: To determine which half-plane to shade, pick a test point not on the boundary line (the origin $(0,0)$ is the simplest choice). If the point satisfies the inequality, shade the region containing that point. If not, shade the opposite side. Visually, this creates a shaded 'zone' of all possible solutions on the coordinate grid.