Inequalities & Modulus

Everything for this chapter, in one place. $x$ is real throughout.

Modulus — the basics


Definition	$\lvert x \rvert = x$ if $x \ge 0$ , and $-x$ if $x < 0$
Always	$\lvert x \rvert \ge 0$ and $\lvert x \rvert^2 = x^2$
Products	$\lvert ab \rvert = \lvert a \rvert \lvert b \rvert$
What it means	$\lvert x - c \rvert$ is the distance from $x$ to $c$

Solving with a modulus


$\lvert x \rvert = a$ , $a > 0$	$x = a$ or $x = -a$
$\lvert x \rvert < a$ , $a>0$	$-a < x < a$
$\lvert x \rvert > a$ , $a>0$	$x < -a$ or $x > a$
$\lvert x \rvert = a$ , $a < 0$	no solution
$\lvert f(x) \rvert = g(x)$	needs $g(x) \ge 0$ — check every root against this

Two moduli


Triangle inequality	$\lvert a+b \rvert \le \lvert a \rvert + \lvert b \rvert$ , equal exactly when $a$ and $b$ have the same sign
Reverse form	$\bigl\lvert \lvert a \rvert - \lvert b \rvert \bigr\rvert \le \lvert a - b \rvert$
Sum of two distances	$\lvert x-p \rvert + \lvert x-q \rvert \ge \lvert p-q \rvert$ , equal exactly when $x$ lies between $p$ and $q$
Worth memorising	$\lvert x+y \rvert + \lvert x-y \rvert = 2\max(\lvert x \rvert, \lvert y \rvert)$

Sign charts — how every rational inequality is solved


Method	factor into linear pieces, mark every root and every break, test one point, alternate
Odd power of a factor	the sign flips across that root
Even power of a factor	the sign does not flip
Denominator roots	always excluded, drawn hollow, even when the inequality is $\le$ or $\ge$
Never do this	cross-multiply by a quantity whose sign you do not know

AM–GM and friends — for positives


Two terms	$\dfrac{a+b}{2} \ge \sqrt{ab}$ , equal only when $a=b$
The workhorse	$t + \dfrac{1}{t} \ge 2$ for $t>0$ , and $\le -2$ for $t<0$
General two-term	$kt + \dfrac{m}{t} \ge 2\sqrt{km}$ for $t>0$
Squares	$x^2 + y^2 \ge 2\lvert xy \rvert$
Fixed sum	the product is largest when the numbers are equal
Fixed product	the sum is smallest when the numbers are equal
Cauchy–Schwarz	$(ax+by)^2 \le (a^2+b^2)(x^2+y^2)$ , equal exactly when $ay = bx$

Quadratic maxima and minima — see the Quadratic chapter


Extreme value of $ax^2+bx+c$	$-\dfrac{D}{4a}$ , reached at $x = -\dfrac{b}{2a}$

Formula sheet