Linear Inequalities

Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. — William Paul Thurston

5.1 Inequalities

Statements $a<b$, $a\le b$, $a>b$, $a\ge b$ are inequalities. A linear inequality in one variable $x$ has the form

$ax+b<0\quad\text{or}\quad ax+b\le 0$

(and similarly $>$ or $\ge$), with $a\mathrel{\htmlClass{course-neq}{\char"2260}} 0$.

5.2 Rules

Adding the same number to both sides preserves the inequality.

Multiplying or dividing by a positive number preserves it.

Multiplying or dividing by a negative number reverses the inequality.

$x>y\ \Rightarrow\ -x<-y$

5.3 Solution on the Number Line

Solve as you would an equation, then mark the solution set on the real line (open/closed endpoints for $<$ vs $\le$).

Interval notation: $x>2$ is $(2,\infty)$; $x\le 3$ is $(-\infty,3]$.

Example: $3x-2\le 7$ $\Rightarrow$ $3x\le 9$ $\Rightarrow$ $x\le 3$, so $(-\infty,3]$.

Two inequalities may be combined: $a\le x\le b$ is a closed interval.

Chapter Summary

• Linear inequality: $ax+b$ compared with $0$
• Reverse the sign when multiplying/dividing by a negative
• Graph the solution on the real line; use intervals

Exercises (NCERT)

• Exercise 5.1 — solve and represent on the number line
• Miscellaneous Exercise