Solving Absolute Value Greater-Than Inequalities
Solving absolute value greater-than inequalities translates |expression| > c into a compound OR inequality: expression > c or expression < -c. This rule applies when the absolute value is isolated and c is positive, and is taught in Openstax Intermediate Algebra 2E, Chapter 2: Solving Linear Equations. The solution is the union of two unbounded intervals on the number line.
Key Concepts
Property When an isolated absolute value inequality uses a "greater than" ($ $) or "greater than or equal to" ($\geq$) symbol with a positive number, it translates into a compound "OR" inequality. If $|u| a$, the equivalent compound inequality is $u < a$ or $u a$. This means the expression must satisfy two completely separate, outward facing conditions.
Examples Simple Greater Than: Solve $|x| 5$. You are looking for numbers whose distance from zero is more than 5. Rewrite as two separate inequalities: $x < 5$ or $x 5$. Multi Step Greater Than: Solve $|2x 1| \geq 7$. Rewrite as two independent inequalities: $2x 1 \leq 7$ or $2x 1 \geq 7$. Solve the first: $2x \leq 6 \rightarrow x \leq 3$. Solve the second: $2x \geq 8 \rightarrow x \geq 4$. The final solution is the union of both: $x \leq 3$ or $x \geq 4$.
Explanation When an absolute value is "greater than" a number, it means the value has escaped the boundaries! It is too far to the left or too far to the right. Because a number cannot be in two completely different places at once, you must split the problem into two entirely separate inequalities connected by the word "or". Solve each one individually to find the two separate rays that make up your final answer.
Common Questions
How do you solve an absolute value greater-than inequality?
Translate |expression| > c into two inequalities: expression > c OR expression < -c, then solve each and take the union.
What type of compound inequality does |x| > c produce?
It produces an OR (union) compound inequality with two separate solution intervals.
Where is solving absolute value greater-than inequalities taught in Openstax?
This is in Openstax Intermediate Algebra 2E, Chapter 2: Solving Linear Equations.
What if c is negative in |expression| > c?
If c is negative, then |expression| > c is always true (absolute value is always non-negative), so the solution is all real numbers.
How is |x| > c different from |x| < c?
|x| > c gives a union of two unbounded intervals (OR), while |x| < c gives one bounded interval (AND).