Introduction:
In 1806, the term ‘modulus’, ‘unit of measure’ was introduced by Jean Robert Argand especially for the complex absolute value. In 1857 onwards the term absolute value was used by French. The absolute value is also called as the ‘numerical value’ or ‘the magnitude’.
What is Absolute value?
The absolute value is defined as the value of real number ‘a’ as its distance from zero or origin which is the numerical value with out sign. The absolute value is also known as modulus and it can be denoted by the notation |a|. Absolute value of any number is always positive or zero but never negative.
Example:
What is the absolute value of -12 and 12?
The absolute value of |12| = 12
|-12| = 12
In various mathematical and physical contexts, the absolute value is closely related to notation of magnitude, distance and norms. Even the complex number, ordered rings, the quaternions, fields, and vector spaces have also the concept of absolute values.
Solving Absolute Value Equations
To solve absolute value equation, the first thing is to convert all negative sign into positive sign. This is the little trick insolving the absolute value equations. The equations are split in to two equations as positive and negative as per the properties of the absolute value. Then the solutions of two equations are found and the solutions are checked by substituting the answer back into the original equation to find out whether the obtained solutions are correct or not.
Solving absolute value inequalities
In solving absolute value inequalities two things such as picture and the absolute value definition should be kept in mind. For example |x| < 3, all the points should be with in 3 and -3 and away from the zero.
The absolute value inequalities can be mentioned as open circle. This equation should be written as 3
Example:
|2x+4| < 6
-6 < 2x+4 <6
In solving absolute value inequalities, the first thing is to spiltthe given equation in to two equations as positive and negative.
2x+4 < 6 2x+4>-6
2x < 2 2x > -10
X < 1 x > -5
So the solution is -5 < x < 1 (the points in between -5 to 1)
How to solve absolute value equations
The steps are:
• Split the given equation into two equations as positive and negative
• Find the solution of two equations
• Substitute the answer back into the original equation to find out whether the solution is correct or not.
Example:
|x+2| = 7
Solving absolute value equation first spilt in to two equations as positive and negative.
X+2 = 7 x+2 = -7
X = 7-2 x = -7-2
X= 5 x = -9
The solutions are = {-9, 5}
Check the answer,
X+2 =7 x+2 = -7
If x=5 if x= -9
5+2 = 7 -9+2 = -7
7 = 7 -7 = -7
