The absolute value of any real number is defined as its overall measured distance from zero on a number line.

If a value ‘x’ represents a real number then the absolute value of ‘x’ is nothing but IxI.

As we see from the definition, every absolute value equation has two solutions. For instance, let us solve for x in the equation, |x| = 12.

Using the definition of absolute value, we know that x is at a distance of 12 units from 0 on the number line. Hence x can be 12 or -12 as both are at a distance of 12 units from 0.

### Absolute Value Definition

The algebraic definition of absolute value, is

If |x| = a if x = 0 and if |x| = -x if x = 0.

Here -x is the opposite of a and not the negative of x.

While solving equations involving absolute values, we must replace the given relationship with two relationships

a) One with the quantity within the absolute value symbols as a positive quantity

b) The other with the quantity within the absolute value symbols as a negative quantity

Once this is done, we must solve the two relationships as any other algebraic equation. For example, the equation |y| = 6 must be replaced as two equations as y = 6 and y = -6. This means one equation, |y| = 6 is equivalent to two equations. Consider another equation involving absolute value |y - 2| = 6. This is equivalent to

y - 2 = 6 OR y - 2 = -6 {Now solve these two equations}

y = 8 OR y = -4

Now let us consider a little more complex situation.