For the sake of simplicity, the following discussion of absolute value equations and inequalities will be restricted to absolute values of linear expressions. Much of this essay is true more generally, but this current discussion is explicitly for Intermediate and College Algebra classes. Extensions can easily be made in other courses when necessary.

Recall that in an earlier discussion of the three siblings it was pointed out that the graph of the equation is a boundary between the graphs of the corresponding inequalities. For that reason, the equation is sometimes called the boundary equation for the inequalities. We will consistently use that terminology.

Recall the definition of absolute value.

Any discussion of equations and inequalities involving absolute values must begin by noting that we are discussing equations and inequalities of the forms

- |M| < k
- |M| = k
- |M| > k

where M is some linear expression and k is a real number.

Observe the Law of Trichotomy dictates that k < 0, k = 0, or k > 0. The three possibilities for k combined with the boundary equation and the two inequalities forces us to consider nine cases. Fortunately, most of these cases are trivial and can be dealt with quite easily. We will separately consider the three possibilities (k = 0, k < 0, and k > 0) for the number k.

Possibility 1: k = 0.

Case 1: (The boundary equation |M| = 0) If M is an algebraic expression, then |M| = k has the form |M| = 0 which is equivalent to M = 0.

To understand this conclusion, look carefully at the definition of absolute value and you will observe that the absolute value of a quantity is 0 if and only if the quantity is 0. Absolute value inequalities of the form |M| = 0 are therefore equivalent to the equality obtained by removing the absolute value symbol. That equality is easily solved using the normal method for solving linear equations in one variable. For anyone who has forgotten this normal method here it is in a nutshell.

The process to solve a linear equation in one variable is to use the following two properties of equations to generate a sequence of equations each equivalent to the previous equation until a simplest equation is obtained.

Properties of Equations:

- If any expression is added to both sides of an equation the resulting equation is equivalent to the original equation.
- If both sides of an equation are multiplied by the same nonzero real number, the resulting equation is equivalent to the original equation.

The following four examples illustrate the solving process when the equality is of the form |M| = 0.

Case 2: (The “less than” inequality |M| < 0) If M is an algebraic expression, then |M| < k has the form |M| < 0 whose solution set is the null set. The absolute value cannot be negative!

Example: |3x – 7| < 0 has no solution. The absolute value of 3x – 7 cannot be negative.

Case 3: (The “greater than” inequality |M| > 0) If M is an algebraic expression, then the solution set for |M| > k has the form |M| > 0 whose solution set is the set of all real numbers except for the solutions of M = 0.

Example: The solution set for |3x – 9| > 0 is all real numbers except 3. The definition of absolute value implies |3x – 9| is positive except when |3x – 9| = 0, which is true if and only if 3x – 9 = 0 which has the solution set {3}. Therefore, the solution set for |3x – 9| > 0 is the set of all real numbers except 3.