Absolute Value, Inequalities, and Intervals

The absolute value of a real number $x$, written $|x|$, is its magnitude irrespective of sign: $|3|=3$ and $|-3|=3$. Geometrically it is the distance from $x$ to $0$ on the real line, and more generally $|a-b|$ is the distance between $a$ and $b$. Inequalities ($<,\le ,>,\ge$) describe sets of numbers rather than single solutions, and these solution sets are most cleanly written as intervals such as $(-2,5]$. Mastering the interplay of these three ideas is essential because most "real" mathematical conditions — tolerances, domains, convergence — are stated as inequalities involving absolute values. A surprising fact: every absolute-value equation or inequality can be solved without ever touching the absolute-value bars, simply by translating it into a statement about distance or into a pair of ordinary inequalities.

The Core Insight

Formal Statement

Absolute Value

For a real number $x$, the absolute value is defined piecewise:

An equivalent algebraic definition uses the square root:

Here $x$ is any real number and $|x|$ is its nonnegative magnitude (no units in pure mathematics; in applications it inherits the unit of $x$).

Interval Notation

Given real numbers $a<b$, intervals denote connected sets of real numbers:

• $(a,b)=\{x:a<x<b\}$ — open, endpoints excluded
• $[a,b]=\{x:a\le x\le b\}$ — closed, endpoints included
• $[a,b)=\{x:a\le x<b\}$ and $(a,b]$ — half-open
• $(a,\infty )=\{x:x>a\}$, $(-\infty ,b]=\{x:x\le b\}$ — unbounded

A round bracket or $\infty$ symbol never includes its endpoint; a square bracket does.

Absolute Value Inequalities

Let $a>0$. Then:

More generally, for a center $c$ and radius $r>0$:

Key Algebraic Properties

For all real $x,y$:

Why It Holds / Derivation

Triangle Inequality

We prove $|x+y|\le |x|+|y|$ for all real $x,y$.

1. Both sides are nonnegative, so the inequality holds iff it holds after squaring.
2. Left side squared: $(x+y{)}^2=x^2+2xy+y^2$.
3. Right side squared: $(|x|+|y|{)}^2=x^2+2|x||y|+y^2$ (using $|x{|}^2=x^2$).
4. Subtracting, the inequality $|x+y{|}^2\le (|x|+|y|{)}^2$ reduces to $2xy\le 2|x||y|$, i.e. $xy\le |xy|$.
5. But $xy\le |xy|$ is always true, since any real number is $\le$ its absolute value.
6. Therefore $|x+y{|}^2\le (|x|+|y|{)}^2$, and taking nonnegative square roots gives $|x+y|\le |x|+|y|$. $■$

The "$<$" Inequality Equivalence, Eq. 3

We show $|x|<a⟺-a<x<a$ for $a>0$.

1. ($\Rightarrow$) Suppose $|x|<a$. If $x\ge 0$ then $|x|=x$, so $x<a$; also $x\ge 0>-a$, giving $-a<x<a$. If $x<0$ then $|x|=-x<a$, so $x>-a$; also $x<0<a$, giving $-a<x<a$.
2. ($\Leftarrow$) Suppose $-a<x<a$. If $x\ge 0$, then $|x|=x<a$. If $x<0$, then $|x|=-x$, and from $-a<x$ we get $-x<a$, so $|x|<a$.
3. Both directions hold, establishing the equivalence. $■$

Karl Weierstrass | 1815–1897

The German analyst Karl Weierstrass introduced the notation $|x|$ for absolute value in 1841 while building the rigorous "epsilon-delta" foundations of calculus. His framework recast continuity and limits in the language $|f(x)-L|<\epsilon$ whenever $|x-c|<\delta$ — making absolute-value inequalities the literal grammar of modern analysis. Weierstrass's insistence on precise inequality-based definitions ended centuries of vague appeals to "infinitely small" quantities.

Building Intuition

Picture the number line as a ruler with $0$ at the center. $|x|$ asks: how many units from the start am I? — and direction is irrelevant. So $|x|=5$ has two answers, $x=5$ and $x=-5$, because two points sit five units from zero.

Now slide the center: $|x-3|$ measures distance from $3$. The inequality $|x-3|<2$ asks for all points within 2 units of 3, which is the interval $(1,5)$ — center $3$, reaching out $2$ in each direction. The "$>$" case flips this: $|x-3|>2$ wants points farther than 2 units from 3, leaving everything outside $(1,5)$, i.e. $(-\infty ,1)\cup (5,\infty )$.

The one habit that prevents nearly all errors: never solve absolute-value problems by guessing — convert to distance or to a double inequality first.

Number-line picture of $|x-3|<2$ vs $|x-3|>2$

Worked Examples

Solving a "less-than" inequality

1. The "$\le$" form means: $-7\le 2x-1\le 7$ (apply Eq. 3 with $\le$).
2. Add $1$ to all three parts: $-6\le 2x\le 8$.
3. Divide by $2$: $-3\le x\le 4$.
4. Endpoints are included because the inequality is non-strict.

Solving a "greater-than" inequality with a tolerance interpretation

1. "Within $0.05$ of $12.00$": $|L-12.00|\le 0.05$ (Eq. 5 with $\le$).
2. Expand: $12.00-0.05\le L\le 12.00+0.05$, so $11.95\le L\le 12.05$.

1. For the failing rod, $|L-12.00|>0.05$ gives $L<11.95$ or $L>12.05$:

Common Misconceptions

Connections

Absolute value and intervals thread through nearly every later topic. The definition $|x|=\sqrt{x^2}$ links to Square Roots and Radicals, and the piecewise definition is a first encounter with Piecewise Functions. Inequality solution sets feed directly into Domain and Range of Functions, since domains are typically described by inequalities. The distance interpretation generalizes to the Distance Formula in the plane and ultimately to Metric Spaces in advanced analysis. Most importantly, the inequality $|f(x)-L|<\epsilon$ is the backbone of Limits and Continuity, where Weierstrass's notation becomes indispensable.

Real-World Applications

1. Manufacturing tolerances. A part specified as $5.00\pm 0.02$ mm is exactly $|L-5.00|\le 0.02$ — quality control is absolute-value inequalities in disguise.
2. GPS and error bounds. A position estimate "accurate to within 3 m" states $|x_{\text{true}}-x_{\text{measured}}|\le 3$.
3. Thermostats and control systems. A heater toggles when $|T-T_{\text{set}}|>\delta$, the deadband interval preventing rapid cycling.
4. Statistics — confidence intervals. A margin of error $\pm E$ around an estimate $\hat{p}$ is the interval $[\hat{p}-E,\hat{p}+E]$.
5. Finance — risk thresholds. A "stop-loss" rule triggers when $|P-P_0|$ exceeds a tolerance, or when price leaves a target band.
6. Signal processing. Amplitude limits $|s(t)|\le A$ prevent clipping in audio and radio transmission.
7. Medicine — therapeutic windows. A drug concentration must stay in an interval $[c_{\min },c_{\max }]$; outside it, the dose is ineffective or toxic.

Historical Development

• 1557 — Robert Recorde introduces the equals sign "$=$" in The Whetstone of Witte, the ancestor of relational symbols.
• 1631 — Thomas Harriot's posthumous work popularizes the inequality signs "$<$" and "$>$".
• 1734 — Pierre Bouguer introduces "$\le$" and "$\ge$" (the symbols with the bar underneath).
• 1821 — Augustin-Louis Cauchy uses inequality-based reasoning extensively in Cours d'analyse, foreshadowing rigorous limits.
• 1841 — Karl Weierstrass introduces the notation $|x|$ for absolute value (then called "numerischer Wert" / numerical value).
• 1854 — The triangle inequality is formalized in the study of metric/normed quantities (geometry context far older, dating to Euclid's Elements, Book I, Prop. 20).
• 1872 — Weierstrass and Heine fully cement the $\epsilon$–$\delta$ definition, making $|f(x)-L|<\epsilon$ standard.
• 1906 — Maurice Fréchet generalizes distance to metric spaces, abstracting $|x-y|$ into a general distance function $d(x,y)$.
• 20th century — Interval notation with round/square brackets becomes the international standard (with the French convention $]a,b[$ for open intervals still in use today).

Subtleties & Limits

The absolute value is not differentiable at $x=0$ — its graph has a sharp corner there, so $|x|$ has no well-defined slope at the origin. This single non-smooth point launches deep questions: optimization with absolute-value (L1) penalties is non-differentiable, requiring subgradient methods rather than ordinary calculus, and these underlie modern techniques like LASSO regression and compressed sensing. Generalizing $|x|$ to a norm $\Vert \mathbf{v}\Vert$ on vectors raises the question of which norms (L1, L2, L∞, ...) best capture "size" — a choice with no universal answer and active consequences in machine learning, where the geometry of the chosen norm shapes which solutions an algorithm prefers. Whether sparse (L1-favoring) representations are always recoverable is governed by subtle conditions still refined in current research.

See Also