Distance and Absolute Value

The Big Idea

On the real number line, absolute value is distance from zero. The distance between any two real numbers is the absolute value of their difference.

\[ |x|=d(x,0), \qquad d(a,b)=|b-a|. \]

Thinking in terms of distance explains absolute-value equations, inequalities, tolerances, errors, deviations, geometric lengths, and many later ideas more reliably than the shortcut “make it positive.”

Distance from the Origin

The points \(5\) and \(-5\) lie on opposite sides of zero, but each is five units from the origin:

\[ |5|=5, \qquad |-5|=5. \]

Absolute value removes directional sign while preserving magnitude. It does not move the original point or claim that \(5=-5\). It reports a separate quantity: distance from zero.

The Piecewise Definition

For every real number \(x\),

\[ |x| = \begin{cases} x, & x\ge0,\\ -x, & x<0. \end{cases} \]

If \(x\) is already nonnegative, its distance from zero is \(x\). If \(x\) is negative, its opposite \(-x\) is positive and gives the distance.

When \(x=-7\), the second case gives

\[ |x|=-x=-(-7)=7. \]

Absolute Value Does Not Simply “Make Positive”

The shortcut works for many nonzero numerical examples but hides important cases and meanings.

• \(|0|=0\), which is nonnegative but not positive.
• If \(x\ge0\), then \(|x|=x\); no sign changes.
• If \(x<0\), then \(|x|=-x\), which is positive.
• Absolute value measures magnitude or distance; it is not an editing command applied to a symbol.

The distance definition continues to work with variables, functions, equations, vectors, errors, and complex numbers.

Basic Absolute-Value Properties

Distance Between Two Number-Line Points

The distance between coordinates \(a\) and \(b\) is

\[ d(a,b)=|b-a|. \]

From \(-4\) to \(7\),

\[ d(-4,7)=|7-(-4)|=|11|=11. \]

Reversing the order produces the same distance:

\[ d(7,-4)=|-4-7|=|-11|=11. \]

What Makes a Distance?

Number-line distance has four important features.

These features become the model for distance functions in geometry, analysis, data science, and other mathematical spaces.

Distance and Directed Change Are Different

Directed change from an initial position \(a\) to a final position \(b\) is

\[ \Delta x=b-a. \]

Its sign records direction. Distance discards that direction:

\[ d=|\Delta x|=|b-a|. \]

Moving from \(6\) to \(-2\) gives \(\Delta x=-8\), while the distance between the endpoints is \(8\). The trip’s path length could be greater than \(8\) if the motion includes detours.

Count Intervals, Not Tick Marks

From coordinate \(2\) to coordinate \(6\), the distance is

\[ |6-2|=4. \]

The visible labels \(2,3,4,5,6\) contain five tick marks but only four unit intervals. Length belongs to the gaps between positions, not to the number of endpoint marks counted.

The scale must also be considered. If adjacent ticks differ by \(5\), crossing three intervals represents a distance of \(15\), not \(3\).

Absolute Value as Center and Radius

The expression

\[ |x-c| \]

measures the distance from \(x\) to the center \(c\). A statement such as

\[ |x-2|=3 \]

asks for points exactly three units from \(2\). There are two:

\[ x=2-3=-1, \qquad x=2+3=5. \]

The center-radius interpretation turns many absolute-value problems into number-line geometry.

Absolute-Value Equations

For \(r>0\),

\[ |u|=r \quad\Longleftrightarrow\quad u=r\ \text{or}\ u=-r. \]

The number of real solutions depends on \(r\):

Solving an Absolute-Value Equation

Solve

\[ |2x-1|=7. \]

Create the two cases:

\[ 2x-1=7 \quad\text{or}\quad 2x-1=-7. \]

Solving gives

\[ x=4 \quad\text{or}\quad x=-3. \]

Both check:

\[ |2(4)-1|=7, \qquad |2(-3)-1|=7. \]

“Less Than” Means Inside a Distance Band

For \(r>0\),

\[ |x-c|<r \quad\Longleftrightarrow\quad c-r<x<c+r. \]

The values lie within \(r\) units of the center \(c\).

For example,

\[ |x-4|<2 \quad\Longleftrightarrow\quad 2<x<6. \]

The inclusive version \(|x-c|\le r\) includes the boundary points \(c-r\) and \(c+r\).

“Greater Than” Means Outside a Distance Band

For \(r>0\),

\[ |x-c|>r \quad\Longleftrightarrow\quad x<c-r\ \text{or}\ x>c+r. \]

The values lie more than \(r\) units from \(c\).

For example,

\[ |x+1|>3 \quad\Longleftrightarrow\quad x<-4\ \text{or}\ x>2. \]

The inclusive form \(\ge\) includes the two boundary points.

Zero and Negative Radius Cases

Distance is never negative, so radius conditions should be inspected before routine manipulation.

Solve the Distance Statement Before Splitting It

Solve

\[ 2|x-3|\le8. \]

Divide by positive \(2\):

\[ |x-3|\le4. \]

Interpret this as points within four units of \(3\):

\[ -1\le x\le7. \]

In interval notation, the solution is \([-1,7]\).

Convert an Interval to Center-Radius Form

For \(a\le b\), the closed interval \([a,b]\) has midpoint

\[ c=\frac{a+b}{2} \]

and radius

\[ r=\frac{b-a}{2}. \]

Therefore

\[ [a,b] = \{x:|x-c|\le r\}. \]

For \([2,10]\), the center is \(6\) and radius is \(4\), so the interval is described by \(|x-6|\le4\).

The Triangle Inequality

For real numbers \(x\) and \(y\),

\[ |x+y|\le|x|+|y|. \]

On a line, traveling directly from \(0\) to \(x+y\) cannot be longer than traveling from \(0\) to \(x\) and then moving by \(y\).

Equality occurs when the two movements point in compatible directions or one is zero. Opposing movements may partially cancel:

\[ |7+(-3)|=4<10=|7|+|-3|. \]

The Reverse Triangle Inequality

The difference between two magnitudes cannot exceed the distance between the original values:

\[ \bigl||x|-|y|\bigr|\le|x-y|. \]

This law provides useful bounds. It says that changing a number by a small amount cannot change its magnitude by more than that amount.

Such bounds become important in approximation, continuity, numerical analysis, and error control.

The Absolute-Value Function

The function

\[ f(x)=|x| \]

folds the negative half of the real line onto the nonnegative half. Its graph is V-shaped, with vertex at \((0,0)\).

A transformed function

\[ f(x)=a|x-h|+k \]

has vertex \((h,k)\). The magnitude \(|a|\) controls vertical scaling, and a negative \(a\) reflects the graph across a horizontal line through its vertex.

Distance in the Coordinate Plane

One-dimensional distance becomes the horizontal component of the two-dimensional distance formula. For points \((x_1,y_1)\) and \((x_2,y_2)\),

\[ d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \]

The differences are directed changes. Squaring removes their signs, and radicalization converts the sum of squared components into ordinary Euclidean length.

In one dimension, this reduces to

\[ \sqrt{(b-a)^2}=|b-a|. \]

Different Problems Can Use Different Distance Rules

Euclidean distance measures straight-line separation. A grid-constrained journey may instead use taxicab distance:

\[ d_{\text{taxicab}} =|x_2-x_1|+|y_2-y_1|. \]

From \((1,2)\) to \((4,6)\), the taxicab distance is \(3+4=7\), while the Euclidean distance is \(5\).

A distance formula should match the movement, geometry, or comparison being modeled.

Magnitude Beyond Real Numbers

For a vector \(\mathbf v=(v_1,v_2)\), Euclidean magnitude is

\[ \|\mathbf v\| =\sqrt{v_1^2+v_2^2}. \]

For a complex number \(z=a+bi\), its modulus is

\[ |z|=\sqrt{a^2+b^2}. \]

Both extend the number-line idea: magnitude describes distance from an origin while omitting direction.

Absolute Error

If \(x_{\text{true}}\) is an accepted value and \(x_{\text{approx}}\) is an approximation, the absolute error is

\[ E_{\text{abs}} = |x_{\text{approx}}-x_{\text{true}}|. \]

If a length is \(10.00\text{ cm}\) and a measurement reports \(9.94\text{ cm}\),

\[ E_{\text{abs}}=|9.94-10.00|=0.06\text{ cm}. \]

The error reports separation, not whether the approximation was above or below.

Relative and Percent Error

Absolute error gains context when compared with the magnitude of the accepted value:

\[ E_{\text{rel}} = \frac{|x_{\text{approx}}-x_{\text{true}}|} {|x_{\text{true}}|}, \qquad x_{\text{true}}\ne0. \]

Percent error is \(100\%\) times the relative error. For the previous measurement,

\[ E_{\text{rel}}=\frac{0.06}{10}=0.006, \qquad E_{\%}=0.6\%. \]

Relative error is undefined by this formula when the accepted value is zero; another scale must then be chosen.

Tolerance as an Absolute-Value Inequality

Suppose a manufactured part should have length \(50\text{ mm}\) within \(0.2\text{ mm}\). If \(L\) is the measured length,

\[ |L-50|\le0.2. \]

Center-radius form immediately gives the acceptable interval:

\[ 49.8\le L\le50.2. \]

Engineering specifications, quality control, calibration, and numerical stopping criteria routinely use this structure.

Deviation from a Reference Value

The expression

\[ |x-c| \]

measures deviation from any reference \(c\), not only from zero.

In statistics, the mean absolute deviation averages distances from a center:

\[ \operatorname{MAD} = \frac1n\sum_{i=1}^{n}|x_i-\bar x|. \]

Positive and negative deviations do not cancel because absolute value records their magnitudes.

Practical Distance Models

If direction matters—early versus late, profit versus loss, above versus below—retain the signed difference alongside its absolute magnitude.

Check with Geometry and Algebra

An absolute-value result should satisfy both an algebraic rule and a distance interpretation.

For \(|x-5|=2\), the algebraic candidates are \(x=7\) and \(x=3\). Geometrically, both points are two units from \(5\):

\[ |7-5|=2, \qquad |3-5|=2. \]

If only one side of the center appears, a case has probably been lost. If a negative distance appears, the interpretation has failed.

A Reliable Distance and Absolute-Value Routine

1. Identify the points or reference value.
2. Decide whether direction matters. Use a signed difference for change and an absolute difference for distance.
3. Check units and scale. Convert compatible units before subtracting.
4. Translate words into center-radius form. “Within,” “at most,” and “no more than” usually signal \(\le\); “more than” signals \(>\).
5. Inspect the radius. Separate positive, zero, and negative cases when necessary.
6. Solve every valid branch.
7. Check the original statement and its geometric meaning.

Common Confusions

Working Summary

Absolute value is distance from zero, and \(|b-a|\) is distance between two real-number points. Distance is nonnegative, symmetric, zero only when the points coincide, and governed by the triangle inequality.

Directed change retains a sign; distance removes it. The expression \(|x-c|\) measures distance from a center \(c\), making absolute-value equations and inequalities center-radius statements about exact, inside, or outside locations.

The same structure supports coordinate geometry, vector magnitude, complex modulus, measurement error, tolerance, deviation, quality control, and other practical models. Always preserve units, domain conditions, and the distinction between magnitude and direction.

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