Absolute Value Inequalities Calculator
Solve inequalities of the form a|bx + c| + d >= e with step-by-step solutions and interval notation
Inequality Input
Current Inequality:
Cannot be zero
Cannot be zero
Any real number
Any real number
Inequality type
Any real number
Solution
Explanation: Solution consists of two unbounded intervals.
Step-by-Step Solution:
Quick Examples
Greater Than
Less Than
Greater Than or Equal
No Solution
Inequality Types
Greater Than
|expr| > k creates OR condition
expr > k OR expr < -k
Less Than
|expr| < k creates AND condition
-k < expr < k
Equal To
|expr| = k has two solutions
expr = k OR expr = -k
Key Properties
|x| >= 0 always
|x| < 0 has no solutions
|x| > negative always true
Dividing by negative flips sign
Solutions often in interval notation
Understanding Absolute Value Inequalities
What are Absolute Value Inequalities?
An absolute value inequality contains an expression within absolute value bars compared to a value using inequality signs (>, <, >=, <=). The general form we solve is a|bx + c| + d ? e, where ? represents any inequality sign.
Solution Method
- Isolate the absolute value expression
- Check the sign of the right side
- Apply the appropriate case rules:
- For > or >=: Create OR conditions
- For < or <=: Create AND conditions
- For =: Create two separate equations
- Solve and express in interval notation
Case Rules
|expression| ? constant
|expr| > k: expr > k OR expr < -k
|expr| < k: -k < expr < k
|expr| = k: expr = k OR expr = -k
Special cases: When k < 0, consider carefully
Real-World Applications
Manufacturing Tolerances
Determining acceptable ranges for product dimensions within specified error bounds.
Temperature Control
Finding temperature ranges that stay within acceptable deviations from target.
Quality Control
Establishing quality ranges where measurements must fall within specified limits.