1.2.1 Single Linear Inequalities¶

Solve and analyze single linear inequalities in one variable, including properties of inequality operations and solution representation.

定义¶

A single linear inequality in one variable is a mathematical statement that compares two linear expressions using inequality symbols: less than ($<$), greater than ($>$), less than or equal to ($\leq$), or greater than or equal to ($\geq$). The general form is $ax + b < c$, $ax + b > c$, $ax + b \leq c$, or $ax + b \geq c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Solving a linear inequality means finding all values of the variable that make the inequality true. The solution set can be represented using interval notation, set-builder notation, or on a number line. Key properties include: (1) Adding or subtracting the same number to both sides preserves the inequality direction; (2) Multiplying or dividing both sides by a positive number preserves the inequality direction; (3) Multiplying or dividing both sides by a negative number reverses the inequality direction.

核心公式¶

$ax + b < c \Rightarrow ax < c - b \Rightarrow x < \frac{c-b}{a}$ (when $a > 0$)
$ax + b > c \Rightarrow ax > c - b \Rightarrow x > \frac{c-b}{a}$ (when $a > 0$)
$ax + b \leq c \Rightarrow x \leq \frac{c-b}{a}$ (when $a > 0$)
$ax + b \geq c \Rightarrow x \geq \frac{c-b}{a}$ (when $a > 0$)
$If $a < 0$: $ax + b < c \Rightarrow x > \frac{c-b}{a}$ (inequality reverses when dividing by negative number)$

易错点¶

⚠️ Forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. For example, solving $-2x + 5 > 9$ incorrectly as $-2x > 4 \Rightarrow x > -2$ instead of the correct $x < -2$.
⚠️ Incorrectly representing the solution set on a number line by using a closed circle instead of an open circle for strict inequalities ($<$ or $>$), or vice versa. For example, using a closed circle for $x < 3$ instead of an open circle.
⚠️ Confusing interval notation with set-builder notation, such as writing $[3, \infty)$ when the correct answer should be $(3, \infty)$ for the inequality $x > 3$, or mixing parentheses and brackets incorrectly.
⚠️ Making algebraic errors when isolating the variable, such as incorrectly combining like terms or making sign errors when moving terms across the inequality sign.