28 Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities is the study of solving equations and inequalities in which the variable appears inside absolute value bars, using a case-splitting technique that accounts for the two possible signs of the expression enclosed by the bars before applying ordinary linear solving methods to each resulting case.

The Scope of Absolute Value Relations

An absolute value equation or absolute value inequality is any equation or inequality in which a variable expression is enclosed within absolute value bars, such as |x - 3| = 7 or |2x + 1| < 9. Because the absolute value function maps both a positive and a negative input to the same non-negative output, solving these relations requires considering two separate possibilities for what the expression inside the bars could equal, rather than proceeding directly as with an ordinary linear equation or inequality.

The Structure of an Absolute Value Equation

An absolute value equation of the form |expression| = k, where k is a non-negative constant, asserts that the enclosed expression has a distance of exactly k from zero. Since exactly two numbers lie a distance k from zero — namely k itself and -k — the enclosed expression must equal one or the other for the equation to be true.

Resolving Absolute Value Equations

Solving |expression| = k, for k > 0, requires splitting the equation into two separate linear equations: expression = k, and expression = -k, solving each independently using ordinary linear equation techniques, and collecting both resulting values as the full solution set.

Solving these two linear equations independently gives x = 10 or x = -4, and both values are genuine solutions to the original absolute value equation, since |10 - 3| = 7 and |-4 - 3| = 7 are both true.

Zero and Negative Equation Cases

Two special cases fall outside the standard two-branch solving pattern. When k = 0, as in |expression| = 0, only a single case applies, since the only number a distance of 0 from zero is 0 itself; the equation reduces to expression = 0, a single linear equation with at most one solution. When k is negative, as in |expression| = -5, the equation has no solution, since an absolute value can never equal a negative number by its own definition, and no algebraic solving steps are needed once this is recognized — the equation is immediately classified as having an empty solution set.

Interior Absolute Value Inequalities

An inequality of the form |expression| < k or |expression| ≤ k (with k > 0) describes values whose distance from zero is less than, or at most, k — meaning the enclosed expression must fall strictly between -k and k. This type of inequality is solved by rewriting it as a chained compound inequality and solving all three parts simultaneously:

The resulting solution set is a single bounded interval, since the condition requires the enclosed expression to stay within a symmetric band around zero.

Exterior Absolute Value Inequalities

An inequality of the form |expression| > k or |expression| ≥ k (with k > 0) describes values whose distance from zero exceeds, or at least reaches, k — meaning the enclosed expression must be either greater than k or less than -k. This type of inequality is solved by splitting into a disjunction of two separate inequalities, joined by "or":

The resulting solution set consists of two unbounded rays, since the condition is satisfied whenever the enclosed expression is sufficiently large in either the positive or negative direction. Distinguishing "less than" absolute value inequalities (which produce a bounded, "and"-joined solution) from "greater than" absolute value inequalities (which produce an unbounded, "or"-joined solution) is essential to selecting the correct case-splitting structure.

Handling Linear Expressions Within the Bars

Regardless of whether the enclosed expression is a simple variable or a more complex linear expression, such as 3x - 5 or -2x + 4, the same case-splitting procedures apply directly to that full expression in place of a bare variable. Each resulting linear equation or inequality, once split, is then solved using the ordinary multi-step or sign-reversal techniques already established for linear equations and inequalities, including careful handling of any negative coefficient on the variable.

Isolating the Absolute Value Before Splitting

When an absolute value equation or inequality is not already presented with the absolute value expression alone on one side, the absolute value portion must first be isolated using ordinary equation or inequality operations before the case-splitting technique is applied. In 3|x - 1| - 4 = 11, adding 4 to both sides and dividing by 3 isolates the absolute value expression to give |x - 1| = 5, which can then be split into its two linear cases.

Verifying Absolute Value Solutions

Every solution obtained from an absolute value equation should be verified by substituting it back into the original, unsplit absolute value equation and confirming the resulting numerical statement is true, since this check also catches any arithmetic errors made while solving either branch. For absolute value inequalities, a solved solution set is verified by testing a value from within the claimed set (confirming it satisfies the original inequality) and a value from outside it (confirming it does not).

Diagnosing Absolute Value Relation Errors

Common errors in this area include solving only one of the two required cases and reporting an incomplete solution set, using the "and" structure for a "greater than" inequality or the "or" structure for a "less than" inequality, forgetting to isolate the absolute value expression before splitting into cases, and failing to recognize immediately that an equation or inequality set equal to or bounded by a negative constant has no solution (in the equation case) or is satisfied by all real numbers (for a "greater than a negative constant" inequality, since absolute value is always non-negative).