Remember that the absolute value of any real number is always positive, or zero (in the unique case of the absolute value of zero itself). So, if you are working with an equation including an absolute value, think of the absolute value as the distance of the number from zero on a simple number line. | 6 | = 6. |-6| = 6. Both 6 and -6 are 6 units away from zero, so they have the same absolute value --- 6.
Perform any arithmetic operations that occur inside or outside the absolute value normally. For example, |4 - 6| = |-2| = 2. 6 - |-4| = 6 - 4 = 2.
Compute the value within the vertical lines, then change the final result to a positive number if it is negative. For example, to determine the solution to |-12 + 4|, don't take the absolute value of -12 and add it to the absolute value of 4. That would be equivalent to |-12| + |4| and equal to 16. |-12 + 4| = |-8| = 8.
Remember that there are two possible values inside the absolute value symbol that will return any given positive value. For example, if |x| = 2, then x = 2 or -2. For a more complex equation, such as |x-5| = 2, assign the expression inside the absolute value symbol to the absolute value, and to the absolute value multiplied by -1. That would yield the two equations: x - 5 = 2 and x - 5 = -2. Compute each equation separately: x = 7 or 3.
Use the formula |a+bi| = Square root (a^2 + b^2) for complex numbers with both a real and imaginary component. This is consistent with the idea that the absolute value is the distance from zero, but instead of the simple real number line, it is the distance from zero on the complex plane, with real numbers on the x axis, and imaginary numbers on the y axis. Again, complex numbers are fairly advanced and students should master working with absolute values for real numbers first. But if they plan on any advanced work in mathematics, science or engineering, they'll need to know about these more complex concepts.