When you hear the term “Absolute Value,” what comes to mind? Isn’t it true that when you think of a number, the first thing that comes to mind is its Magnitude? The absolute value represents only the numeric value and does not include the sign of the numeric value. For example, to describe distance, we don’t use positive and negative integers; instead, we utilise direction and its absolute value. In practice, “absolute value” means that any negative sign in front of a number should be removed, and all numbers should be thought of as positive (or zero).

It is essential for students to learn these important concepts in Mathematics as it would make the basic conceptual knowledge stronger in them. Let’s take a closer look at the absolute value, read on.

What is Absolute Value?

Absolute value describes how far a number is from \(0\) on the number line without considering direction. We all know that distance is a positive number. Because the absolute value is a measure of distance, it is never negative. In addition to the value, a sign is sometimes assigned to a numeric value to indicate the direction.

A positive or negative value is occasionally assigned to a numeric value to explain an increase or decrease in quantity, values above or below the mean value, profit, or loss in a transaction. The sign of the numeric value is ignored in absolute value, and only the numeric value is examined.
The Absolute Value is denoted by two vertical lines like this \(|\,|.\) The absolute value of the number \(7,\) for example, is expressed as \(\left| 7 \right| = 7.\) This means that the distance from \(0\) is \(7\) units.

Similarly, \(\left| { – 7} \right| = 7\) denotes the absolute value of a negative \(7.\) This means that the distance from \(0\) is \(7\) units. The number line isn’t simply for displaying the distance from zero; it may also be used to graph equalities and inequalities containing absolute value expressions.

Take a look at the equation \(\left| x \right| = 4.\) To display \(x\) on the number line, you must display all numbers with an absolute value of \(4.\) There are just two spots in the world where this occurs at \(4\) and at \( – 4\)

Example of Absolute Value

The absolute value of \(x\) is the distance of \(x\) from \(0,\) indicated by \(\left| x \right|\) (and interpreted as “the absolute value of \(\left| x \right|\)”). As a result, the absolute value can never be negative.
Let’s see some typical examples which involve Absolute value

1. Evaluate \( – \left| { – 7} \right|\)
Given \( – \left| { – 7} \right|,\) we first solve the absolute value part and then we take the negative of that number
Hence, \( – \left| { – 7} \right| = – \left( { + 7} \right) = – 7\)

2. Evaluate \( – \left| {2 + 3 – 7} \right|\)
Given \( – \left| {2 + 3 – 7} \right|,\) we first simplify the content inside the absolute value part and then we take the negative of that number
Hence, \( – \left| {2 + 3 – 7} \right| = – \left| { – 2} \right| = -2\)

3. Evaluate \({\left( { – \left| { – 4} \right|} \right)^2}\)
\({\left( { – \left| { – 4} \right|} \right)^2} = {\left( { – \left( 4 \right)} \right)^2} = 16\)

Function of Absolute Value or Absolute Value Function

The function \(f:R \to {R^ + } \cup 0\) defined by \(f\left( x \right) = \left| x \right| = \left\{ {\begin{array}{*{20}{c}} {x,}&{x \ge 0}\\ { – x,}&{x < 0} \end{array}} \right.,\) is the called modulus function. It is denoted with the symbol \(|\,|.\) It is also called an Absolute value function.

Its domain is \(R\) (set of real numbers) and range is \({R^ + } \cup 0,\) i.e. positive real numbers including \(0.\)

When dealing with variables, it is difficult to determine the sign of the number or the value contained within that variable. For example, given the variable \(x,\) you can’t know if it contains a \(2\) or a \( – 4\) merely by looking at it. What would you say if I asked you for the absolute value of \(x?\) Because it’s impossible to discern if a variable has a positive or negative value merely by looking at it.

We’d have to think about these two scenarios separately. When you take the absolute value of \(x > 0\) (that is, if \(x\) is positive or zero), the value does not change. If \(x = 2,\) for example, you have \(\left| x \right| = \left| 2 \right| = 2 = x.\) In fact, the sign would be constant for any positive value of \(x\) or if \(x = 0\) so:
\(\left| x \right| = x\) for \(x > 0\)

When you take the absolute value of \(x,\) it will change its sign if \(x < 0\) (that is, if \(x\) is negative). If \(x = – 4,\) for example, \(\left| x \right| = \left| { – 4} \right| = + 4 = – \left( 4 \right) = – x,\) In reality, the sign would have to be modified for any negative value of \(x\) so:
\(\left| x \right| = – x\) for \(x < 0\)

This is an example where the minus sign on the variable indicates a change of sign from whatever the sign originally was, rather than a number to the left of zero. This “–” does not signify “the number is negative”, but rather the sign on the original value has been modified.
For example: \(\left| {3.5} \right| = 3.5\)
\(\left| { – 3.7} \right| = – \left( { – 3.7} \right) = 3.7\)

Absolute Value Equations

The absolute value is used to determine the numeric value of a quantity, regardless of its sign. Many quantities, such as length, price, and volume, have no meaning for the sign and are written without one. To represent such amounts, the concept of absolute value is useful.
Let’s see some equations which involve the Absolute value.
If \(\left| {x + 5} \right| = 20\) then \(\left( {\left( {x + 5} \right) = 20\,{\rm{or}}\left( {x + 5} \right) = – 20} \right) \Rightarrow x = – 15,\, – 25\)
If \(\left| {x – 5} \right| = 20\) then \(\left( {\left( {x – 5} \right) = 20\,{\rm{or}}\left( {x – 5} \right) = – 20} \right) \Rightarrow x = 25,\, – 15\)
If \(\left| {3 – x} \right| = 20\) then \(\left( {\left( {3 – x} \right) = 20\,{\rm{or}}\left( {3 – x} \right) = – 20} \right) \Rightarrow x = – 17,\, – 23\)
If \(\left| {x + 5} \right| = – 2\) then \(x = \emptyset \) as No value of \(x\) will satisfy this equation and number from the absolute value part can’t be negative.
If \(\left| {\frac{x}{9}} \right| = 4\) then \(\left| {\frac{x}{9}} \right| = 4\) or \(\left| {\frac{x}{9}} \right| = – 4 \Rightarrow x = 36,\, – 36\)
If \(\left| {\frac{9}{x}} \right| = 0\) then \(x = \emptyset \) as No value of \(x\) will satisfy this equation

What are the Absolute Value Properties?

The Absolute value properties help in solving linear equations and many real-life word problems. These properties are as follows:

If \(a\) and \(b\) is any real numbers, then the absolute values are satisfying the following properties,

Solved Examples: Absolute Value

Q.1. Find all the possible values of \(x\) which satisfy \(\left| {2x} \right| = 16\)
Ans: Given, \(\left| {2x} \right| = 16\)
\( \Rightarrow 2x = \pm 16\)
\( \Rightarrow 2x = 16\) or \(2x = – 16\)
\( \Rightarrow x = 8\) or \(x = – 8\)

Q.2. Simplify \( – \left| { – 8 + 4 – 3} \right|\)
Ans: Given, \( – \left| { – 8 + 4 – 3} \right|\)
\( \Rightarrow – \left| { – 11 + 4} \right|\)
\( \Rightarrow – \left| { – 7} \right|\)
\( \Rightarrow – 7\)

Q.3. A sea diver is \( 60\) feet beneath the water’s surface. To reach the surface, how far must he swim?
Ans: He must swim a distance of \(\left| { 60} \right| = 60\) feet

Q.4. What is the absolute value of \(20 – 36\left( 2 \right) + 2\left( {40 – 37} \right)\)?
Ans: Given, \(\left| {20 – 36\left( 2 \right) + 2\left( {40 – 37} \right)} \right|\)
\( \Rightarrow \left| {20 – 72 + 6} \right|\)
\( \Rightarrow | – 46| = 46\)

Q.5. Solve the equation given below
\(\left| {x + 4} \right| = 5\)
Ans: \(\left| {x + 4} \right| = 5\)
\( \Rightarrow x + 4 = \pm 5\)
\( \Rightarrow x + 4 = 5\) or \(x + 4 = – 5\)
\( \Rightarrow x = 1\) or \(x = – 9\)

Q.6. Solve the equation \(\left| x \right| = x + 2.\)
Ans: \(\left| x \right| = x + 2\)
\( \Rightarrow x = \pm \left( {x + 2} \right)\)
\( \Rightarrow x = x + 2\) or \( \Rightarrow x = – x – 2\)
\( \Rightarrow x = – 1\)

Summary

In this article, we have learnt about the absolute number, examples and also their properties and got to know that it plays an essential role in the mathematical world. It is used to determine the numeric value of a quantity, regardless of its sign. The absolute value of a number can never be negative. This concept is widely used to solve mathematical equations.

Frequently Asked Questions

Q.1. What is an absolute value of \(10?\)
Ans: The absolute value of the number \(10\) is \(10\) since \(\left| {10} \right| = 10.\)
The absolute value of a number \(x,\) in algebraic terms, takes \(x\) and makes it positive.

Q.2. Explain absolute value in math.
Ans: Absolute value describes how far a number is from \(0\) on the number line without considering direction. We all know that distance is a positive number. Because the absolute value is a measure of distance, it is never negative. The sign of the numeric value is ignored in absolute value, and only the numeric value is examined.

Q.3. What is an absolute value of \(\left| { – 7} \right|?\)
Ans: The absolute value of the number \( – 7\) is \(7\) since \(\left| { – 7} \right| = 7.\)
The absolute value of a number \(x,\) in algebraic terms, takes \(x\) and makes it positive.

Q.4. What is an absolute value of \(\left| { – 4} \right|?\)
Ans: The absolute value of the number \( – 4\) is \(4\) since \(\left| { – 4} \right| = 4.\)
The absolute value of a number \(x,\) in algebraic terms, takes \(x\) and makes it positive.

Q.5. How do you teach absolute value?
Ans: Absolute value refers to a number’s distance from zero on the number line without taking direction into account. A number’s absolute value can never be negative. For example – Walking \(100\) feet west is equivalent to walking \( + 100\) feet, and walking \(100\) feet east is equivalent to walking \( – 100\) feet. We don’t use positive and negative integers to describe distance; instead, we use direction and its absolute value. And in another example, A sea diver is \( – 60\) feet beneath the water’s surface. To reach the surface, He must swim a distance of \(\left| { – 60} \right| = 60\) feet.

Q.6. Why is the absolute value necessary?
Ans: It is because sometimes we just need to use positive values, and the absolute value is a handy tool for this. If you see an absolute value in a problem or equation, it signifies that everything inside it is always positive. Absolute values are frequently utilised in distance problems and are occasionally utilised with inequalities.

Q.7. What is absolute value of a number?
Ans: Absolute value refers to a number’s distance from zero on the number line without taking direction into account. A number’s absolute value is never negative.

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