U23 Linear Inequalities in One Unknown: Essential Formulas

Definition and Standard Form

A linear inequality in one unknown $x$ can be expressed in the form $ax + b > 0$, $ax + b < 0$, $ax + b \ge 0$, or $ax + b \le 0$, where $a$ and $b$ are real numbers and $a \neq 0$.

Addition/Subtraction Rule

Adding or subtracting the same number from both sides of an inequality does not change the inequality direction. If $a > b$, then $a + c > b + c$ and $a - c > b - c$ for any real number $c$.

Multiplication/Division Rule (Critical)

Multiplying or dividing both sides by a positive number preserves the inequality sign. Multiplying or dividing by a negative number reverses the inequality sign.

General Solution Steps

To solve $ax + b \ \square \ 0$ ($\square$ represents $>$, $<$, $\ge$, or $\le$): 1. Isolate the $x$ term. 2. Divide by the coefficient $a$, remembering to reverse the inequality if $a < 0$. 3. Express the solution set.

where $\square'$ is the original or reversed inequality sign based on the sign of $a$.

Representing Solutions

Solutions are typically represented in one of three ways: using inequality notation, set-builder notation, or graphically on a number line.