text: # Absolute Value – Definition, Function, Symbol, Examples, Facts, FAQs
  
  ## What Is the Absolute Value of a Number?
  
  The absolute value of a number tells us how far a given number is from the origin on a number line.
  
  The absolute value only talks about distance, and not direction. We know that distance is a non-negative quantity. Thus, the absolute value of a number can never be negative.
  
  So, what does absolute value mean? Observe the diagram. Here, we can see that
  
  7 is 7 units to the right of 0.
  
  Thus, the absolute value of 7 is 7.
  
  -7 is 7 units to the left of 0.
  
  Thus, the absolute value of -7 is 7.
  
  ## Absolute Value: Definition
  
  Absolute value of a number can be defined as the distance of the number from the origin on a number line.
  
  ## Absolute Value: Symbol
  
  Absolute value of a number is represented by writing the number between two vertical bars. Note that the vertical bars are not to be confused with parentheses or brackets
  
  The absolute value of x is represented by |x|, and we read it as “absolute value of x.” It is also read as “modulus of x.” Sometimes, the absolute value of a number n is also written using the notion “abs(n).”
  
  In the previous example, we can write
  
  $|7| = | \;–\; 7| = 7$
  
  ## How to Find the Absolute Value of a Number
  
  The absolute value of a positive number is the number itself. The absolute value of a negative number is written as the numeric value ignoring the negative sign. The absolute value of 0 is 0.
  
  ## Absolute Value of 0
  
  The distance of the point zero from zero is 0 itself. Thus, the absolute value of 0 is 0.
  
  ## Absolute Value Properties
  
  Non negativity: Absolute value of a number is always non-negative.
  
  $| n | \ge 0$
  
  Symmetry: The absolute value of a number equals the absolute value of its additive inverse.
  
  $| \;-\; x | = | x |$
  
  Positive definiteness: Absolute value of a number is 0 only if the number is 0.
  
  $| n | = 0$ if and only if $n = 0$
  
  On the same lines, we can say that $| x \;–\; y | = 0 ↔ x = y$
  
  Multiplicativity: The absolute value of the product of two numbers is the product of the absolute values of the individual numbers.
  
  $|x\times y| = | \times | \times | y |$
  
  Preservation of division (equivalent to multiplicativity):
  
  $|\frac{x}{y}| = |\frac{x}{y}| …y \neq 0$
  
  Sub-additivity: The absolute value of the sum of two numbers is less than or equal to the addition of the absolute values of the individual numbers.
  
  $|x + y| \le | x | + | y |$
  
  Equivalent to sub-additivity: $| x \;–\; y | \ge | | x | \;–\; | y | |$
  
  Triangle inequality (equivalent to subadditivity): $| x \;–\; y | \le | x \;–\; z | + | z \;–\; x |$
  
  Idempotence: The absolute value of an absolute value of a number is simply the absolute value.
  
  $| | x | | = | x |$
  
  ## Absolute Value Inequalities
  
  Let’s discuss two important absolute value inequalities.
  
  1. $| x | \le a ⇔ \;−\; a \le x \le a$
  
  This is often referred to as AND inequality.
  
  Example: $|x \;−\; 5| \le 7 ⇔ \;−\;7 \le x \;−\; 5 \le 7$
  
  $⇔ \;−\; 2 \le x \le 12$
  
  1. $| x | \ge a ⇔ x \le \;−\; a\; or\; x \ge a$
  
  This is often referred to as OR inequality.
  
  Example: $| x | \ge 3 ⇔ x \le \;−\;3$ or $x \ge 3$
  
  ## Absolute Value of Real Numbers
  
  The absolute value of a positive real number is the number itself. The absolute value of a negative real number is the number without a negative sign. The absolute value of 0 is 0.
  
  Thus, for a real number x, the absolute value is given by:
  
  $| x | = x$ if $x \ge 0$
  
  $| x | = \;–\; x$ if $x \lt 0$
  
  ## Absolute Value on a Number Line
  
  As mentioned earlier, the absolute value of a number tells us the distance of a number from zero on a number line.
  
  It is easy to understand the absolute value of integers (or any number) using the number line.
  
  Example: The absolute value of 8 is 8.
  
  $| 8 | = 8$.
  
  Note that the absolute value of $\;–\; 8$ is also 8.
  
  $| \;–\; 8 | = 8$.
  
  ## Absolute Value of a Number Examples
  
  • $| 1 | = 1$
  • $| \;-\;1| = 1$
  • $| 3.5 | = 3.5$
  • $| 0 | = 0$
  • $| 7.9 | = 7.9$
  • $| 5 \;–\; 3 | = | 2 | = 2$
  • $| \;–\; 1 + 6 | = | 5 | = 5$
  • $| \;–\; 6 \times 2| = | \;–\; 12 | = 12$
  
  ## Absolute Value Function
  
  The absolute value function can be given by f(x) = |x| such that:
  
  • $| x | = + x$  for $x \gt 0$
  • $| x | = \;–\; x$  for  $x \lt 0$
  
  ## Applications of Absolute Value
  
  • To find the distance between two points.
  
  The absolute value of the difference of two real numbers (absolute difference) is the distance between them.
  
  • To identify and remove outliers in data analysis.
  • To set tolerance in quality analysis.
  • To determine the degree of variation in the data compared from a certain point.
  
  • The absolute value of x, denoted by | x |, is also known as “modulus” of x.
  • $| x | \lt$ a implies that  $\;-\;a \lt x \lt a$
  • $| x | \gt$ a implies that $x\lt (\;-\;a)$ or $x \gt a$
  
  ## Conclusion
  
  In this article, we learned about the absolute value of a number, its symbol, properties, absolute value function, and we also learned its applications. Let’s move ahead and solve a few examples and MCQs based on the concept of absolute value for practice.
  
  ## Solved Examples on Absolute Value
  
  1. What is the absolute value of  $\;–\;18$?
  
  Solution:
  
  The absolute value of a negative number is written as the numeric value, ignoring the negative sign.
  
  The absolute value of $\;-\;18 = |\;-\;18| = 18$
  
  2. Find the value of $-|\frac{\;-\;15}{22}|$.
  
  Solution:
  
  First, focus on the absolute value part.
  
  The absolute value of $\;-\;\frac{15}{22} = |\frac{-15}{22}| = |\frac{15}{22}|$
  
  There’s a minus sign outside the absolute value. It will get tagged along with the answer.
  
  $-|\frac{-15}{22}| = \frac{-15}{22}$
  
  3. Find: $|-9| + |10|$.
  
  Solution:
  
  Here,
  
  |\;-\; 9 = 9|
  
  $|10| = 10$
  
  Thus, $|\;-\;9| + |10| = 9 + 10 = 19$
  
  4. Find the absolute value.
  
  $|5|, |\;-\;11|, |0|, |\;-\;7|, |\;-\;10|$
  
  Solution:
  
  $|5| = 5$
  
  $|\;-\;11| = 11$
  
  $|0| = 0$
  
  $|\;-\;7| = 7$
  
  $|\;-\;10| = 10$
  
  5. Simplify: $|x \;−\; 1| \le 2$
  
  Solution:
  
  $|x \;−\; 1| \le 2$
  
  $\;−\;2 \le x \;−\; 1 \le 2$
  
  $(\;−\;2 + 1) \le x \le (2 + 1)$
  
  $\;−\; 1 \le x \le 3$
  
  ## Practice Problems on Absolute Value
  
  1
  
  ### The absolute value of $\;–\; 25$ is ___.
  
  $\;–\; 25$
  25
  0
  50
  CorrectIncorrect
  $|\;–\; 25| = 25$
  2
  
  ### $|11| \times | \;-\; 12| =$ ___
  
  $\;–\; 132$
  0
  132
  $\;–\; 264$
  CorrectIncorrect
  $|11| = 11$
  $|\;-\;12| = 12$
  $|11| \times | \;-\;12| = 11 \times 12 = 132$
  3
  
  ### Find the value of $|\frac{-3}{5}| \times |\frac{5}{6}|$
  
  $\;– \;\frac{1}{2}$
  0
  1
  $\frac{1}{2}$
  CorrectIncorrect
  Correct answer is: $\frac{1}{2}$
  $|\frac{-3}{5}| \times |\frac{5}{6}| = \frac{3}{5} \times \frac{5}{6} = \frac{1}{2}$
  4
  
  ### $|x \;-\; y| = 0$ if and only if
  
  $x = \;-\; y$
  $x = y$
  $x \neq y$
  $xy = 0$
  CorrectIncorrect
  Correct answer is: $x = y$
  $|x\;-\;y| = 0$ if and only if $x = y$.
  5
  
  ### $| 0 | =$
  
  0
  1
  $\;-\;1$
  Not defined
  CorrectIncorrect
  Absolute value of 0 is 0.
  6
  
  ### The absolute value of 8 is the distance between ____ on a number line.
  
  8 and -8
  8 and 1
  8 and 0
  -8 and 1
  CorrectIncorrect
  Correct answer is: 8 and 0
  The absolute value of 8 is the distance between 8 and 0 on a number line.
  
  $|\;-\;x| = x$
  
  So, $-\; |\;-\;x| = \;-\; x$
  
  $| x | \le$ a can be written as a compound inequality $\;−\; a \le x \le a$.
  
  $| x | \ge a$ can be simplified as $x \le \;−\; a$ or $x \ge a$.
  
  Example: $| x | \ge 1$ means $x \le \;−\;1$ or $x \ge 1$.
  
  Yes, two different numbers can have the same absolute value. The two numbers $\;–\;10$, and 10 have the same absolute value of 10.
  $| \;–\; 10 | = | 10 | = 10$.
  
  The absolute value of 0 is always 0 as the distance of 0 from 0 is 0.
  
  Absolute numbers or absolute values simply refer to numeric values, ignoring the sign of the given numbers.
  
  Absolute value is a distance, and distance can never be negative. Thus, the absolute value is always positive. To be more accurate, the absolute value is a non-negative number. Absolute value of 0 is 0. 0 is neither positive nor negative. It is considered a non-negative as well as a non-positive integer.
  
  We use a bar on either side of a number to mean absolute value. The absolute value bars or the vertical bars | | represent “modulus symbol.” It gives us the distance of the number from 0.
  
  Absolute value is also commonly referred to as numerical value or magnitude.