Calculator Inputs
This form solves inequalities of the form |ax + b| relation c.
Example Data Table
This sample shows how common absolute value inequalities map to interval notation.
| Example | Equivalent form | Solution |
|---|---|---|
| |x - 3| < 5 | -5 < x - 3 < 5 | (-2, 8) |
| |2x + 4| ≤ 6 | -6 ≤ 2x + 4 ≤ 6 | [-5, 1] |
| |x + 1| > 3 | x + 1 < -3 or x + 1 > 3 | (-∞, -4) ∪ (2, ∞) |
| |3x - 9| ≥ 0 | Always true | (-∞, ∞) |
| |x - 7| = 2 | x - 7 = -2 or x - 7 = 2 | {5} ∪ {9} |
Formula Used
Standard rules
1. For |u| < c, rewrite as -c < u < c.
2. For |u| ≤ c, rewrite as -c ≤ u ≤ c.
3. For |u| > c, rewrite as u < -c or u > c.
4. For |u| ≥ c, rewrite as u ≤ -c or u ≥ c.
5. For |u| = c, rewrite as u = -c or u = c.
In this calculator, u = ax + b. After rewriting the absolute value statement, the page solves the resulting linear inequalities and returns interval notation.
Important edge cases are included automatically. For example, |u| < 0 has no solution, while |u| ≥ 0 is true for all real numbers.
How to Use This Calculator
- Enter the coefficient a and constant b for the inner expression
ax + b. - Choose the relation symbol:
<,≤,>,≥, or=. - Enter the right-side constant c.
- Set your preferred decimal precision and variable label.
- Click Solve Inequality to display the result above the form.
- Review the interval notation, graph, step-by-step method, and downloadable table.
FAQs
1. What does an absolute value inequality represent?
It describes all x-values whose distance-based expression satisfies a comparison. Absolute value measures distance from zero, so solutions often become intervals or unions of intervals.
2. Why do some answers show two intervals?
Expressions like |u| > c or |u| ≥ c split into two cases: one left of -c and one right of c. That creates a union of intervals.
3. What happens when the right side is negative?
Because absolute value is never negative, inequalities such as |u| ≤ -2 have no real solution. Inequalities like |u| > -2 are true for every real number.
4. Why is interval notation useful?
Interval notation summarizes solution sets compactly. It clearly shows endpoints, inclusion, exclusion, and separated regions without repeating multiple inequality statements.
5. Can this solve an absolute value equation too?
Yes. Choose the equals sign. The calculator converts |u| = c into u = -c or u = c, then returns the point solutions.
6. Why do I sometimes get all real numbers?
Some statements are always true. For example, |u| ≥ 0 holds for every real value because absolute value cannot be below zero.
7. What does the graph help me see?
The graph compares |ax + b| with the horizontal target line. Where the curve is below, above, or touching that line matches the solution rule you selected.
8. Can I export the calculated values?
Yes. The page includes CSV and PDF export buttons for the computed check table, which makes classroom review and documentation easier.