Theory

Equations and inequalities with absolute value

The piecewise definition of absolute value, its properties, and the systems that solve equations and inequalities with one or more absolute values.

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Definition and geometric meaning
Properties

Definition and geometric meaning

1. Absolute value

The absolute value of a real number $x$ is

∣ x ∣ = {x - x if x \geq 0 if x < 0

The result is never negative: when $x < 0$ , the number $- x$ is positive. For example $∣7∣ = 7$ and $∣ - 7∣ = - (- 7) = 7$ . Every solving rule in this chapter follows from this piecewise definition.

y = \left|x\right|

y = |x|: for x ≥ 0 it coincides with y = x, for x < 0 with y = −x

2. Absolute value as distance

For $x, c \in R$ , the number $∣ x - c ∣$ is the distance between $x$ and $c$ on the real line; in particular $∣ x ∣$ is the distance of $x$ from the origin. This reading translates equations and inequalities into conditions on points of the line and makes it possible to check the results. For example $∣ x - 3∣ = 2$ is satisfied by the points at distance $2$ from $3$ , that is $x = 1$ and $x = 5$ .

Properties

Properties of absolute value

For all $a, b \in R$ ,

∣ ab ∣ = ∣ a ∣ ∣ b ∣, ∣ a ∣^{2} = a^{2}, ∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣.

The first is verified by examining the four cases for the signs of $a$ and $b$ , and it is used to split an absolute value, for example $∣ - 3 x ∣ = 3 ∣ x ∣$ . The second holds because $a$ and $∣ a ∣$ differ at most in sign, which squaring cancels; since absolute values are not negative, comparing two absolute values is equivalent to comparing their squares, and this observation is the basis for solving $∣ A ∣ > ∣ B ∣$ by squaring. The third is the triangle inequality and is used in estimates, for example in the chapter on limits: for $a = 5$ and $b = - 3$ we have $∣5 + (- 3) ∣ = 2 \leq 8 = ∣5∣ + ∣ - 3∣$ , and equality holds when $a$ and $b$ have the same sign.

Warning

The absolute value is not removed by rewriting $∣ A ∣ = A$ : the equality holds only where $A \geq 0$ . For example $∣ x - 5∣ = x - 5$ is false at $x = 0$ , since $∣0 - 5∣ = 5$ while $0 - 5 = - 5$ . Where $A < 0$ , instead, $∣ A ∣ = - A$ .

The theory continues with 7 more sections (the equation a k, the equation a b, the equation a b 1…), plus all worked examples, the solutions and the PDF.

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