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Absolute Value Equations

Study material notes on Absolute Value Equations, history of Absolute Value, thorough explanation of Absolute Value Equations, and other related topics.

An absolute value equation is an equation in which the variable is present within the absolute value operator. The typical example of it is |x-5|=9. The signs of the numbers greatly define their absolute value. If it is positive, the number will equal |9|=9. If the number is negative, the absolute value will be drawn out as |-9|=9.

This article talks about the absolute value equation. You will find brief information on the concept of absolute value equation and absolute value in Maths, a thorough explanation of absolute value symbol, its properties, and ways to solve the absolute value equation. So, let’s start by describing the absolute value equation in the Maths study material.

History of Absolute Value 

The term absolute value was first coined in 1806 when Jean-Robert Argand introduced the term module for representing the complicated absolute value. Later, the spelling changed to modulus. Although the term modulus is still used occasionally, absolute value and magnitude are interchangeable.

Explain Absolute Value Equations

The absolute value of a number is its distance from 0. Since we know that the distance is usually a positive number and the absolute value can be described as the measure of distance, we can say that it is never negative. 

A negative or positive value is assigned occasionally to the numeric value for explaining a rise or fall in the quantity, values above or below the mean value, profit, or loss in a transaction.

The formula for calculating the absolute value – 

Explain absolute value symbol 

We write a vertical bar on either side of a number (or a variable) to denote its absolute value, i.e., |x|, where x is an integer. For example, the absolute value of 2 is expressed as |2|. In addition, the absolute value of -2 is denoted by the symbol |-2|. As previously stated, the absolute value always yields a non-negative value. As a result, |2|=|-2| = 2. That is, it converts negative integers to positive numbers as well.

Properties of absolute value 

There are four fundamental qualities of absolute value. These include Multiplicativity, Non-negativity, subadditivity and positive-definiteness. While these appear intimidating, it is easier to understand them through examples – 

  • |a| = 0 ⇔ a = 0: Positive-definiteness denotes that a number’s absolute value is zero only if the number is zero.

  •  |ab| = |a| |b|: The absolute value of a product of two numbers equals the product of the absolute values of each number, which is known as multiplication. For example, |(3)(-4)| = |3| |-4| =(3)(4) = 12.

  • |a| ≥ 0: non-negativity refers to a number’s absolute value being larger than or = to zero.

  • |a + b| ≤ |a| + |b|: The absolute value of the sum of two real numbers is less than or equal to the total of the absolute values of the two numbers, according to subadditivity. For example, |2 + -3| ≤ |2| + |-3| because 1 ≤ 5.

The identity of the indiscernible, idempotence, symmetry, triangle inequality and division preservation are all significant features.

  • ||a|| = |a|: The absolute value of the absolute value is the absolute value, according to idempotence.

  • |-a| = |a|: The absolute value of a negative number is the same as the absolute value of its positive value, according to symmetry.

  • |a – b| = 0 ⇔ a = b: Positive-definiteness is an analogous phrase for identity of indiscernible. When both a and b have the same value, the absolute value of a – b is zero.

  •  |a – b| ≤ |a – c| + |c – b|: Subadditivity is equivalent to the triangle of inequality.

  • |a / b| = |a| / |b| if b ≠ 0: Multiplicativity is the same as division preservation.

How to solve absolute value equations 

Solving the absolute value equation is quite easy. Please note that the absolute value of a positive and negative integer is the same. For building the right expressions, use the following points of the absolute value – 

  • The first step is to separate the absolute value expression from the rest of the expression.

  • The next step is solving the expression in absolute value notation so that it can be expressed as a positive (+) or negative (-) quantity.

  • It’s time to find a solution for the unknown.

  • In order to double-check your work, you can either graph it or insert the answers into the equation.

Conclusion 

With this, we come to an end to absolute value equations. The absolute value of a number is its distance from zero (0). Since we know that the distance is usually a positive number and the absolute value is a measure of distance, we can say that it can never be negative. 

In this article describing absolute value equations, we studied the concept of absolute value equations and absolute value in length. We covered several other topics, such as the history of absolute value, properties of absolute value, ways to solve absolute value equations, and other related topics. We hope this study material helped you better understand absolute value equations.

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