Absolute Value

A number's absolute value is its magnitude regardless of its sign. The absolute value has no sign; regardless the number is positive or negative. It indicates the distance of the number from $0$ and distance can never be negative and so absolute value.

Absolute Value Definition

For any real number, the absolute value or modulus of a number is denoted by $\left|x\right|$, with a vertical bar on each side of the quantity, and is defined as

$$\begin{gathered} \left|x\right|=\left\{\begin{array}{l}x if x>0\\ 0 if x=0 \\ -x if x<0\end{array}\right. \end{gathered}$$



The absolute value of $x$ is thus always either a positive number or zero, but never negative. When $x$ is positive or $0$, the absolute value is the number itself. When $x$ is negative, the absolute value is negative of that $x$, so the result is ultimately positive.

For example, 
$$\begin{gathered} \left|5\right|=5 \\ \left|0\right|=0 \\ |-5|=-(-5)=5 \end{gathered}$$

Absolute Value as a Distance from a Point

The absolute value of a real number is that number's distance from zero along the real number line. The absolute value of the difference between two real numbers is their distance. As the distance can never be negative, the absolute value can also never be negative.

This can be pictorially represented as follows:



The absolute value of $4$ is the distance of $4$ from $0$.




The absolute value of $-5$ is the distance of $-5$ from $0$.

Points Representing $\mathbf{\left|}\mathit{x}\mathbf{\right|}\mathbf{=}\mathit{a}$

Since $\left|x\right|$ is equal to the distance of $x$ from the origin, we are looking for those numbers that are lying at a units of distance from zero. We can see from the figure given below that the distance of both the points $a$ and $-a$ is a units from $0$. Hence, the equation $\left|x\right|=a$ represents the points $x=a$ and $x=-a$.



For example, $\left|x\right|=2$ represents the points $x=2$ and $x=-2$ and can be represented as

Points Representing $\mathbf{\left|}\mathit{x}\mathbf{\right|}\mathbf{<}\mathit{a}$

$\left|x\right|=a$ represents the distance of point $a$ or $-a$ from $0$. Hence, $\left|x\right|<a$ represents all the points whose distance from $0$ is less than $a$ units. All these points can be represented as below:


The thick green region represents the points belonging to $\left|x\right|<a$. Hence, the points representing the inequation $\left|x\right|<a$ can be defined by the inequality $-a<x<a$.

For example, $\left|x\right|<2$ can be written as $-2<x<2$ and can be represented as

Absolute Value to Find Distance Between Two Numbers

$|x-a|$ represents the distance of point $x$ from point a. Similarly, $|x+a|$ represents the distance of the number $x$ from the number $-a$ as $|x+a|=|x-(-a\left)\right|$. So, when we write the equation $|x-a|=p$, it denotes that the distance between $x$ and $a$ is $p$ units. That is, we are looking for points whose distance from $a$ is $p$ units. This can be represented as



For example, $|x-4|=5$ means the point $x$ and $-x$ that is $5$ units away from $4$. We can represent this as


$|x-4|=5$ means $x=9$ or $x=-1$.

Region Representing $\mathbf{|}\mathit{x}\mathbf{-}\mathit{a}\mathbf{|}\mathbf{<}\mathit{p}$

$|x-a|=p$ gives two points for $x$ that satisfies the equation. Whereas the inequality $|x-a|<p$ represents a region of points for which $|x-a|$ is less than $p$. In this region, the distance of the point $a$ from $x$ will be less than $p$. This can be represented as


The thick green line represents the points of $x$ that satisfy the inequality $|x-a|<p$.

For example, $|x-4|<5$ can be represented as below.


The region can be represented by an open bracket $(-1,9)$ as the endpoints are not valid for the inequality.

Now, what about $|x-a|>p$? This will be the region outside the boundary of $|x-a|<p$, and it can be shown as




 The region shown in a thick green line represents the values satisfying the inequality $|x-a|>p$.  The rest of the region of $|x-a|<p$ represents $|x-a|>p$.

For example,

$|x-4|<3$ is represented by the green region, and $|x-4|>3$ is represented by red.

$\mathbf{|}\mathit{x}\mathbf{-}\mathit{a}\mathbf{|}\mathbf{+}\mathbf{|}\mathit{x}\mathbf{-}\mathit{b}\mathbf{|}$

$|x-a|$ and $|x-b|$ are the distances of numbers $a$ and $b$ from $x$. When $x$ is lying between $a$ and $b$, the sum of the distances of $x$ from $a$ and $b$ is constant and is equal to the distance between $a$ and $b$, i.e., $|a-b|$. When $x$ lies on either side of $a$ and $b$, the sum $|x-a|+|x-b|$ starts increasing as $x$ starts moving further away.

Case 1: $x$ is lying between $a$ and $b$




Case 2: $x$ lies on either side of $a$ and $b$



For example,

Solved Examples

Q1. Find the value of $x$ for which $\mathbf{|}\mathit{x}\mathbf{-}\mathbf{5}\mathbf{|}\mathbf{=}\mathbf{4}$.
Ans: $|x-a|=p$ give two points for $x$ that satisfy the equation. We need to find the value of $x$ such that it is $4$ units away from $5$.



So, $x=1$ and $x=9$.

Q2. Find the value of $\mathit{x}$ for which $\mathbf{|}\mathit{x}\mathbf{-}\mathbf{2}\mathbf{|}\mathbf{+}\mathbf{|}\mathit{x}\mathbf{+}\mathbf{3}\mathbf{|}\mathbf{=}\mathbf{8}$.

Ans: $|x-2|+|x+3|$ is the sum of distances of the number $x$ from $2$ and $-3$, respectively. Here, $x$ can have two cases,
Case 1: $x$ is between $2$ and $-3$
If $x$ lies between $2$ and $-3$, then $|x-2|+|x+3|=2-(-3)=5$. As there are only $5$ points between $2$ and $-3$, $x$ between them is impossible.

Case 2: $x$ lie on either side of $2$ and $-3$ at a distance $c$ and $5+c$ from two numbers as shown below: 



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Hence, the value of $x$ are $2+1.5=3.5 \mathrm{or} -3-1.5=-4.5$

Q3. Find the range where $\mathbf{|}\mathit{x}\mathbf{-}\mathbf{2}\mathbf{|}\mathbf{>}\mathbf{3}$.
Ans: Let us first identify the range where $|x-2|<3$. This can be denoted as $-1<x<5$.



All the values outside this range will satisfy the inequality $|x-2|>3$. It can be denoted as $x<-1$ and $x>5$ and represented as

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