# Solve Absolute Value Inequalities

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Absolute value of any number whether it’s negative or positive represents only its positive version. Absolute value is actually the distance of the number from the center and is denoted by “| |“(modulus sign). Hence absolute value is never negative. Absolute value is the magnitude of the number not the sign. Absolute value equation contains the variables and numbers with the absolute value sign in it.

Example 1: Solve for the variable x in the inequality | 2x | > 4?
Solution: Given is the equation |2x| = 4.
Here the unknown variable which needs to be solved for is x.
First step: The absolute value gives two cases; 2x > 4; 2x < -4.
Dividing by 2 on both sides of the given equation.
(2x)/ 2 > 4/2; (2x)/ 2 < -4/ 2
This gives x > 2; x < -2.

Hence the solution contains two intervals x > 2 and x < -2.

Example 2: Solve for the variable x in the inequality | 3x + 1| > 10?
Solution: Given is the equation |3x + 1| > 10.
Here the unknown variable which needs to be solved for is x.
First step: The absolute value gives two cases; 3x + 1 > 10; 3x + 1 < -10.
Subtracting 1 on both sides. 3x > 9; 3x < -11.
Dividing by 3 on both sides of the given equation, x > 9/3; x > -11/3
This gives x > 3; x < -11/3.

Hence the solution contains two intervals x > 3 and x < -11/3.