Linear Inequalities - Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation

A linear inequality in one variable is an algebraic expression involving a single variable with the highest power of 1, connected by inequality signs such as $<$, $>$, $\le$, or $\ge$. For example, $ax + b < 0$ is a linear inequality where $a \neq 0$.

Solving an inequality involves finding all real numbers that make the statement true. Unlike equations that typically have one solution, inequalities often result in an infinite set of values known as the 'solution set'.

Rule of Addition/Subtraction: Adding or subtracting the same number to both sides of an inequality does not change the direction of the inequality sign. For instance, if $x - 3 > 5$, then $x > 8$.

Rule of Multiplication/Division: Multiplying or dividing both sides by a positive number preserves the inequality sign. However, multiplying or dividing by a negative number reverses the direction of the inequality (e.g., if $-2x < 10$, then $x > -5$).

Graphical Representation on a Number Line: The solution set is visualized as a shaded region on a horizontal number line. A 'strict' inequality ($<$ or $>$) is represented by an open circle ($\circ$) at the endpoint to show it is excluded. A 'slack' inequality ($\le$ or $\ge$) is represented by a solid/dark circle ($\bullet$) to show the endpoint is included.

Interval Notation: Solutions are often written in interval form. Open intervals $(a, b)$ use parentheses for excluded endpoints, while closed intervals $[a, b]$ use square brackets for included endpoints. Infinity ($\infty$) and negative infinity ($-\infty$) are always paired with parentheses.

Compound Inequalities: When a variable is bounded by two values, such as $a < x \le b$, the graphical representation is a line segment between $a$ and $b$, with an open circle at $a$ and a dark circle at $b$.