The absolute value is defined as the distance of ‘a’ from zero and is denoted by the symbol |a|. The absolute value only tells how far from zero and not in which direction is the location of a.
From the above figure, the absolute value of |3| = 3 since it is 3 units from the right side of the zero, and |-3|= 3 since it is 3 unit from the left side of the Zero. So the absolute unit should not have the negative sign, and is always positive or zero.
Absolute value equation
To solve absolute value equations, first split the equation into two equations. Then solve the equation to get the solution of absolute value equation.
Example 1:
|x| = 7
X = 7 x = -7
The solutions are {7, -7}
Example2:
|3x-4| = 5
3x-4 = 5 3x-4 = -5
3x = 9 3x = -1
X = 3 x = -1/3
The solutions are {1/3, 3}
Absolute Value Inequality
To solve absolute value inequalities first isolate the one side on the inequality symbol. Then write the first equation with out absolute sign and solve the inequalities and write second equation without absolute sign, reverse the inequalities and then solve the problem. The absolute value should be greater than any negative number and it should not be less than a negative number, sincethe absolute value should be always positive.
Example1: (greater than)
|x-20| > 5
x-20> 5 X-20 < -5
x> 25 x < 15
The solution is 15 > x > 25
Example2: (less then or equal to)
|X-3| = 4
X-3 = 4 X-3 = -4
X = 7 X = -1
The solution is -1 = x = 4
Absolute value of a complex number
The definition for absolute value ofa real number is not directly generalized for definition for complex numbers, since the complex numbers are not ordered. But the geometric interpretation of the absolute value of a real number is as its distance from ‘0’ to be generalized.
The definition for absolute value ofa complex number is, it is a distance in the complex plane from the origin ‘0’ by using of Pythagorean Theorem. The absolute value of the difference of the complex number is equal to the difference between them.
The definition for absolute value ofa complex number is,
Z = x + iy
Where,
Z - Absolute value or modulus
x, y - Real numbers
|z| = v(x^2+y^2 )
In above equation, the absolute value of a real number is x, when the complex part becomes 0.
In polar form the complex number z is expressed like z = re i?
If the absolute value is (r = 0, ?-real)
|z| = r
So, the absolute value of a complex number in the complex analogue equation is,
The properties of absolute value of a complex number are as same as absolute values of a real numbers.
Give example how to solve the absolute value equations and inequalities
Example 1: absolute value equation
|x - 7| = |2x-2|
Write in to two equations with out absolute symbol.
X-7 = 2x-2 x-7 = -(2x-2)
X-2x = -2+7 x+2x = 7 + 2
-x = 5 3x = 9
X = -5 x = 9
The solutions are { 9, -5}
Example 2: Absolute value inequalities
|2x-3| > 5
Write in to two inequalities with out absolute symbol.
2x-3 > 5 2x-3 < - 5
2x > 8 2x< -2
X < 4 x < -1
The solution is -1 > x < 4
