Absolute Value - Guide On The Fundamental Concept

Need To Know About Absolute Value: 101 Math Guide

Have you been asked to find the absolute value of |3x|=9 by your professor as soon you enter the class? Are you racking your brains to figure out how to solve this problem accurately? Believe us; you are not the only one.

Absolute value is an overly tricky subject. Most students constantly struggle to find the absolute value in a numerical or graphical approach. However, the absolute value is one of the most significant concepts when learning math for any student. Individuals can never progress if they haven't quite mastered absolute value as a basic concept.

In today’s comprehensive post, we will walk you through an ultimate guide of absolute value. If you were always striving to understand such a key component of mathematics, then go through this post diligently to unravel certain crucial information you must know to have a brilliant command over place value in the long run.

Let’s dive right in!

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Introduction to Absolute Value: A Quick Overview

The absolute value in mathematics is defined as the value that describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute of a number can never have negative values.

Take a look at some examples-

The absolute value of 5 is 5. The distance from 5 to 0 is 5 units.

The absolute value of -5 is 5. The distance from -5 to 0 is 5 units.

Now, the absolute value can be explored both numerically and graphically. Let's understand each one of them-

Absolute Value- The Numeric Approach

Numerically, the absolute value is often found to be indicated by enclosing a number, variable, or expression inside two vertical bars, like so-

‘|20|

|x|

|4n-9|

Now, it is wise not to confuse the |absolute value bars| with (parentheses) or [brackets]. They are not the same symbols, and different rules apply for calculating them.

Like, -1(-3)=3. The negatives inside and outside the parentheses cancel out when they are multiplied. But -1|-3|=-3. It is impossible to multiply across the absolute value bars, so it is crucial to first find the absolute value of the numbers inside them. Since the absolute value of -3 is 3, the operation becomes -1(+3).

When the absolute value bars enclose an expression that incorporates operations, the expression must be assessed before finding the absolute value. For instance, take the expression |6-4|. Before we take the absolute value of the expression, it is vital to carry out the subtraction. When you do that, |6-4| becomes |2|. Now, the absolute value of the expression can be found – it's the absolute value of 2, which is 2.

So, |6-4|=|2|=2

Absolute Value- The Graphic Approach

On a number line, the absolute value of a number or expression is the distance between the value and zero. When using a number line to explore absolute value, the absolute value of an expression will always lie at zero or to the right of zero. If the original value is positive or zero, the absolute value will overlie the original.

If we plot both the original and absolute values, they will be in the same place. The |3| is 3. In this case, both the absolute and original value will be like 3 units to the right of zero on a number line.

If the original value is negative, the absolute value will lie at the same distance from zero as the original but on the other side of the origin. The |-4| is 4. If we plot both absolute and original values, they are the same distance as zero but in opposite directions.

When using the number line to show the absolute value of an expression, it is crucial to make sure to carry out all the operations inside the bars before plotting. The |4-6| is plotted at 2, not -2,4, or 6.

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Absolute Value of An Integer

The absolute value of an integer implies its numerical value without considering the sign. It is the distance from zero on the number line. The absolute value of +3 is 3, and the absolute value of -3 is 3. Hence, opposite integers have the same absolute value.

The symbol used to represent the absolute value is two vertical lines (||), one on either side of the integer. Thus, if 'a' represents an integer, its absolute value is represented by |a| and is always non-negative.

Examples-

The absolute value of -7 is written as |-7|= 7 [here mod of -7=7]
The absolute value of +2 is written as |+2|=2 [here mod of +2=2]

Absolute Value of A Real Number

The modulus or absolute value of a real number x denoted as |x| is the non-negative value of x about its sign.

So, |x|= x if x is a positive number and |x|=-x if x is negative.

For example, the absolute value of the real number 3 is 3, and the absolute value of -3 is also 3. The absolute value of a real number may be thought of as its distance from zero.

Absolute Value Properties

The absolute value has the below-enlisted four fundamental properties-

Non-negativity = |a|≥0
Positive-definiteness = |a|=0⇔a=0
Multiplicativeness = |ab|=|a||b|
Subaddivity = |a+b|≤|a|+|b|

Other important properties of the absolute value include-

Idempotence = ||a||=|a|
Symmetry = |−a|=|a|
Identity of indiscernibles= |a−b|=0⇔a=b
Triangle inequality= |a−b|≤|a−c|+|c−b|
Preservation of division= |a/b|=|a|/|b|if b≠0

Absolute Value Function

An absolute function can be defined as a function that contains an algebraic expression within absolute value symbols.

The absolute value parent function, written as f(x)= |x| is defined as,

f(x)= {x if x> 0

{0 if x=0

{-x if x<0

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How To Find The Absolute Value Of A Number?

The absolute value of a number is incredibly easy to find. This theory behind it is important when solving absolute value equations. Take a look at the below-enlisted steps to find the absolute value of a number accurately and effectively-

Know the absolute value is a number’s distance from zero

An absolute value is mainly the distance from the number to zero along a number line. Simply put, |-4| is just asking how far away -4 is from zero. Since the distance is always a positive number, the result of the absolute value is always positive.

Make the number in the absolute value sign positive

At its simplest, absolute value makes any number positive.

Use simple and vertical bars to show the absolute value

The notation for the absolute value is easy. Single bars around a number or expression like |n|, |3+5|, or |-72| demonstrates absolute value.

Drop any negative number inside the absolute value marks

Like, |-5| would become |5|.

Drop the absolute value marks.

The number remaining is your answer, so |-5| becomes |5| and then 5.

Simplify the expression inside the absolute value sign

If you’ve got a simple expression like |-10|, you can simply make the whole thing positive. However, the expressions like |(-4*5) + 3-2| must be simplified before you can take the absolute value. The normal order of operations still applies-

Problem : |(4*5) +|3-2|
Simplify inside parenthesis: |(-20) + 3-2|
Add and Subtract- |-19|
Make everything inside the absolute value positive: |19|
Final Answer: |19|

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Absolute Value Examples

Find the possible values of x in |4x|=16

In this equation, 4x can either be positive or negative. So, we can write it as-

4x=16 or -4x=16

Divide both sides by 4

X=4 or x=-4

Hence, the two possible values of x are -4 and 4.

Find the absolute value of a number -12/5

To find |-12/5|

|-12/5|= 12/5

Hence, the absolute value of a number is -12/5

Anna wants to find the following values- a. |-13/5| b. -|-3|, c. |2(-3)+|4|

Solution-

The absolute value results in non-negative values all the time. Hence, we have the following solutions-

|-13/15|= 13/15
-|-3|= -(3)= -3
|2 (-3) +4| = |-6+4|= |-2|= 2

Mastering essential math concepts like absolute value in the early years is the key to success in the classroom.

Save this post on the laptop. Take the help of this guide and develop a lucid understanding of the absolute value to become a confident solver in the long run. Break a leg!