Last updated: July 31, 2026
Absolute Value Inequalities Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
a|bx + c| + d ? e ⇒ |bx + c| ? (e-d)/a after normalization
Where:
- a,b,c,d,e=Coefficients and constants
- ?=One of >, ≥, <, ≤
Worked Examples
Outside solution with ≥
Solve |x-3| ≥ 5.
- 1Normalize to |x-3| ≥ 5
- 2Split: x-3 ≤ -5 or x-3 ≥ 5
- 3Solve each inequality
Inside solution with <
Solve |2x| < 3.
- 1Normalize to |2x| < 3
- 2Write double inequality -3 < 2x < 3
- 3Divide by 2
Always true case
Solve |x| > -1.
- 1Absolute value is always ≥ 0
- 20 > -1 is always true
- 3All x satisfy the inequality
Introduction
This calculator solves linear absolute-value inequalities and returns both solution wording and interval notation, including all-real and no-solution edge cases.
Formula and Method
The Absolute Value Inequalities Calculator uses the core relation a|bx + c| + d ? e ⇒ |bx + c| ? (e-d)/a after normalization.
Apply the formula directly to user inputs
Compute intermediate terms when needed
Round numeric outputs for stable display
Return structured outputs for UI rendering
Input Fields
Provide values for: a, b, c, d, operator, e.
- 1
a: a
- 2
b: b
- 3
c: c
- 4
d: d
- 5
operator: Inequality operator
- 6
e: e
Output Fields
The calculator returns: solutionSet, intervalNotation, normalizedTarget, inequalityType, boundary1, boundary2.
- 1
solutionSet: Human-readable x-solution
- 2
intervalNotation: Set notation in interval form
- 3
normalizedTarget: Solved target k in |bx+c| ? k
- 4
inequalityType: Outside, inside, all real, no solution, or invalid
- 5
boundary1: First boundary value
- 6
boundary2: Second boundary value
Validation Rules
Inputs must be numeric and operator must be one of >, ≥, <, ≤. The solver handles special cases when a=0 or b=0.
Invalid or missing numeric values trigger safe fallback outputs
Error messaging is returned by the calculation function when constraints fail
Domain restrictions are enforced before formula evaluation
Rounding is applied to final numeric values
How to Use This Calculator
Enter valid inputs, run the calculation, and review both primary and supporting outputs.
Enter each required field
Click calculate
Check primary output first
Use examples to verify expected behavior
Practical Uses
This calculation is commonly used in math practice, engineering checks, and fast verification tasks.
Homework and exam preparation
Quick sanity checks while modeling
Teaching and tutoring demonstrations
Reusable reference for manual calculations
FAQs
What does the Absolute Value Inequalities Calculator compute?
It computes solutionSet, intervalNotation, normalizedTarget, inequalityType, boundary1, boundary2 from the provided inputs.
Which inputs are required?
Required fields are a, b, c, d, operator, e.
How are results rounded?
Numeric outputs are rounded by the calculation function (generally to about 6 decimal places) for cleaner display.
What if I enter invalid values?
Inputs must be numeric and operator must be one of >, ≥, <, ≤. The solver handles special cases when a=0 or b=0.
Which output should I read first?
Start with solutionSet, then use the remaining outputs for context and interpretation.
Can I use decimal values?
Yes. Decimal inputs are supported unless the calculator specifically requires integer-style values.