Calculator Input
Example Data Table
| Example | Input | Type | Solution |
|---|---|---|---|
| 1 | |2x - 6| = 8 | Equation | x = 7, -1 |
| 2 | |3x + 9| < 6 | Strict inequality | -5 < x < -1 |
| 3 | |x - 4| ≤ 2 | Closed interval | 2 ≤ x ≤ 6 |
| 4 | |4x + 8| ≥ 12 | Outside interval | x ≤ -5 or x ≥ 1 |
Formula Used
For a linear absolute value expression, start with |ax + b|. The definition splits into two branches:
- |u| = u when u ≥ 0
- |u| = -u when u < 0
For equations, |ax + b| = c becomes ax + b = c and ax + b = -c.
For inequalities:
- |ax + b| < c means -c < ax + b < c
- |ax + b| ≤ c means -c ≤ ax + b ≤ c
- |ax + b| > c means ax + b < -c or ax + b > c
- |ax + b| ≥ c means ax + b ≤ -c or ax + b ≥ c
How to Use This Calculator
- Enter coefficient a from the linear expression inside the bars.
- Enter constant b from the same inside expression.
- Choose the relation: equation, less than, greater than, or inclusive versions.
- Enter the right side constant c.
- Set the variable symbol if you prefer something other than x.
- Click Solve Now to view steps, interval notation, and the graph.
- Use the export buttons to save your result summary.
Frequently Asked Questions
1. What does this solver handle?
It solves linear absolute value equations and inequalities of the form |ax + b| relation c. It returns interval notation, branch logic, turning point information, and a graph.
2. Why can some problems have no solution?
Absolute value is never negative. Because of that, equations like |ax + b| = -3 and inequalities like |ax + b| < 0 cannot be true for real numbers.
3. Why do some equations give two answers?
The same positive distance can come from two opposite linear values. So |u| = c usually splits into u = c and u = -c, which often produces two roots.
4. What does the turning point mean?
The turning point is where the inside expression equals zero. On the graph, that is the lowest point of the V-shape. It helps identify symmetry and interval boundaries.
5. How are strict and inclusive inequalities different?
Strict inequalities use open interval endpoints because boundary values are excluded. Inclusive inequalities use closed endpoints because the boundary values satisfy the relation.
6. Does the sign of a change the result?
Yes. The sign of a affects root order and the inside condition, though the absolute value graph still looks like a V-shape. The solver automatically accounts for this.
7. Can I use decimals?
Yes. The calculator accepts decimal coefficients and constants. Results are rounded for readability while preserving the actual calculation structure used in solving.
8. What do the export buttons save?
The CSV button downloads a compact summary table. The PDF button opens the browser print dialog, which you can save as a PDF from your device.