Solving Absolute Value Equations and Inequalities Cheat Sheet

Key Facts

• Absolute value means distance from zero, so $\left|x\right| \ge 0$ for every real number $x$.
• For $\left|A\right| = k$, if $k > 0$, solve the two equations $A = k$ and $A = -k$.
• For $\left|A\right| = 0$, solve only $A = 0$ because zero has no positive and negative pair.
• For $\left|A\right| = k$, if $k < 0$, there is no solution because an absolute value cannot be negative.
• For $\left|A\right| < k$ with $k > 0$, write the compound inequality $-k < A < k$.
• For $\left|A\right| \le k$ with $k \ge 0$, write the compound inequality $-k \le A \le k$.
• For $\left|A\right| > k$ with $k \ge 0$, write the compound inequality $A < -k$ or $A > k$.
• For $\left|A\right| \ge k$ with $k > 0$, write the compound inequality $A \le -k$ or $A \ge k$.

Vocabulary

Absolute value

The absolute value $\left|x\right|$ is the distance of $x$ from $0$ on the number line.

Compound inequality

A compound inequality combines two inequalities using an idea like "and" or "or" to describe a solution set.

Solution set

The solution set is the collection of all values that make an equation or inequality true.

Boundary point

A boundary point is a value where an inequality changes from true to false or false to true.

Extraneous solution

An extraneous solution is a value found during solving that does not satisfy the original equation or inequality.

Isolate

To isolate an expression means to use inverse operations until that expression is alone on one side of the equation or inequality.