Enter Inequality Values
This solver handles inequalities in the form |ax + b| operator c.
Plotly Graph
The chart compares y = |ax + b| with the horizontal line y = c.
Example Data Table
| Inequality | Key Idea | Solution |
|---|---|---|
| |2x - 3| ≤ 5 | Middle region with endpoints included | [-1, 4] |
| |x + 4| > 2 | Outside region with endpoints excluded | (-∞, -6) ∪ (-2, ∞) |
| |3x| < 0 | Absolute value cannot be less than zero | ∅ |
| |x - 1| ≥ 0 | Absolute values are always nonnegative | (-∞, ∞) |
| |5x + 10| ≤ 0 | Must equal zero exactly | {-2} |
Formula Used
Case 1: Less Than or Less Than Equal To
For |ax + b| < c or |ax + b| ≤ c, rewrite it as -c < ax + b < c or -c ≤ ax + b ≤ c. This gives the middle interval.
Case 2: Greater Than or Greater Than Equal To
For |ax + b| > c or |ax + b| ≥ c, rewrite it as ax + b < -c or ax + b > c. This creates two outer solution rays.
Boundary Point Formula
Boundary points come from |ax + b| = c. Solve ax + b = c and ax + b = -c. That gives x = (c - b) / a and x = (-c - b) / a.
Special Cases
If c is negative, some inequalities have no solution while others include all real numbers. If c = 0, the result may become one point or every real value except one point.
How to Use This Calculator
- Enter the coefficient a from the linear expression inside the absolute value.
- Enter the constant b so the inside becomes ax + b.
- Choose the inequality operator: <, ≤, >, or ≥.
- Enter the right side constant c.
- Pick a variable name and display precision if needed.
- Press Solve Inequality to show the result below the header and above the form.
- Review interval notation, set notation, boundary points, and the step explanation.
- Use the CSV or PDF buttons to export the current solved output.
Frequently Asked Questions
1) What form does this solver handle?
It solves inequalities of the form |ax + b| < c, |ax + b| ≤ c, |ax + b| > c, and |ax + b| ≥ c. Decimals, negatives, and zero values are all accepted.
2) Why can the answer contain two intervals?
When the operator is > or ≥, values far enough from the center satisfy the inequality. That creates two separate rays instead of one middle interval.
3) What happens when c is negative?
Absolute values are never negative. Because of that, a negative right side creates special cases. Some operators give no solution, while others make every real number a solution.
4) Why does c = 0 matter so much?
Zero can turn the result into a single point, all real numbers, or every value except one point. The exact outcome depends on the operator you choose.
5) Do the boundary points belong to the answer?
Use ≤ or ≥ to include boundary points. Use < or > to exclude them. The interval notation shows this clearly with brackets or parentheses.
6) How does the graph help me understand the answer?
The graph compares y = |ax + b| with y = c. Their intersection points mark boundaries, and the satisfied regions explain the interval solution visually.
7) Can I export my solved result?
Yes. Use the CSV button for spreadsheet-friendly output or the PDF button for a clean printable summary of the current inequality result.
8) Does the solver support decimal inputs?
Yes. Decimal coefficients and constants are supported. Adjust the precision setting to control displayed rounding for interval endpoints, boundary points, and summary values.