Algebraic Solutions of Linear Inequalities in One Variable

Solving a linear inequality algebraically means performing a sequence of legal operations on both sides until the variable stands alone, exactly as with an equation — but two of the rules behave differently here, and getting them right is the single most important skill in this chapter.

Rule 1 — Adding or subtracting the same quantity. If $a < b$, then for any real number $c$,

Adding or subtracting the same number on both sides never changes the direction of the inequality (and the same holds for $\le$, $>$, $\ge$).

Rule 2 — Multiplying or dividing by a positive number. If $a < b$ and $k > 0$, then

Multiplying or dividing both sides by a positive number also preserves the direction.

Rule 3 — Multiplying or dividing by a negative number. If $a < b$ and $k < 0$, then

Why does the direction reverse? (Proof.) Suppose $a < b$, so $b - a > 0$. Let $k < 0$; then $-k > 0$. Multiplying the positive number $b-a$ by the positive number $-k$ keeps the product positive:

So the flip is not an arbitrary rule but a direct consequence of "positive times positive is positive." A quick numerical check confirms it: $2 < 5$ is true, but multiplying both sides by $-1$ gives $-2$ and $-5$, and indeed $-2 > -5$ — the number that was smaller became the one that is less negative, i.e. larger.

General method. To solve a linear inequality in one variable: (1) simplify both sides — clear brackets, combine like terms; (2) collect every variable term on one side and every constant on the other, using Rule 1 (this step never needs a flip); (3) if the final coefficient of the variable is negative, divide by it using Rule 3 and reverse the sign; if it is positive, use Rule 2 and keep the sign as is.

Worked illustration. Solve $8 - 3x < 2$ for real $x$. Subtracting 8 from both sides (Rule 1): $-3x < -6$. Dividing both sides by $-3$ — a negative number — reverses the inequality by Rule 3: $x > 2$. The solution set is $\{x \in \mathbb{R} : x > 2\}$, the interval $(2, \infty)$. …