## Pages

### Simple Absolute value equations

Simple absolute value equations are of the form |3x + 2| = 1. They contain an expression inside an absolute value on one side and a number on the other.

Simple absolute value equations have different solutions depending on whether the number on the right hand side is positive, negative or zero.

If the number is positive, as in |3x + 2| = 1, the absolute value can be removed by taking a negative and a positive value of the number. From |3x + 2| = 1,
• Take positive value of 1: 3x + 2 = 1
• Take negative value of 1: 3x + 2 = -1
Solve both the equations to obtain two different values of 'x'. For the above example, the solutions are given below

 Fig. 1
The two values of x obtained from solving |3x + 2| = 1 are -1/3 an -1. These are the solutions to the equation.

If the number is zero:
Absolute value equations in which the right hand side number is zero have only one solution, since the negative and positive values of zero are equal. For example, the equation |3x + 2| = 0 has only one solution, that obtained by solving 3x + 2 = 0 for 'x'.

If |3x + 2| = 0 is solved by the method as outlined in the above example (Fig. 1), two equations are obtained,
• 3x + 2 = 0
• 3x + 2  = -0
Since 0 is same as -0, hence the two equations are same. There is no need to both equations separately. The solution can be obtained by solving 3x + 2 = 0:
3x + 2 = 0
3x = -2
x = -2/3
Thus x = -2/3 is the solution to the absolute value equation |3x + 2| = 0.

If the number is negative:
Any expression inside an absolute value can have a negative or positive value, but its absolute value is always a positive value. Thus, if in an absolute value equation, the number on the right hand side is negative, the equation is false for all real values of 'x'. In other words, absolute value equations having a negative number on the right hand side have no solution.

For example, the equation |3x + 2| = -1 has no solution.

So in order to solve simple absolute value equations, first check whether the number on the right hand side is positive, negative or zero, and proceed accordingly, as follows:
• If the number is positive, get two equations, one with the positive value of the number and the other with the negative value of the number
• If the number is negative, there is no solution
• If the number is zero, there is only one solution, which is obtained by solving the equation without absolute value signs.