What this calculator does
Absolute value inequalities works out solves |ax + b| compared to c (<, ≤, >, ≥) and shows the solution in interval notation. Enter your own figures above and the answer updates as you type: nothing is fixed in the code, so the result reflects exactly the numbers you supply.
The formula this calculator evaluates is printed under the tool and explained below, so you can check the working by hand or reuse it in a spreadsheet.
The formula
Formula|ax + b| < c ⟺ −c < ax + b < c (and similarly for ≤, >, ≥, flipping to a union of rays for > and ≥)
The inputs explained
| Field | What to enter |
|---|---|
| a (coefficient of x) | A number. Starts at 2. |
| b (constant inside) | A number. Starts at -3. |
| Compared to | Choose from < c, ≤ c, > c, ≥ c. |
| c | A number. Starts at 5. |
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the answer changes with a (coefficient of x)
Every other input is held at the calculator’s starting values while a (coefficient of x) varies. Select any row to load that scenario into the calculator.
How the answer changes with a (coefficient of x)
| a (coefficient of x) | Solution set | Boundary values | Formula used |
|---|---|---|---|
| 1 | (-2.000, 8.000) | x = -2.000 and x = 8.000 | |ax + b| compared to c, solved from −c < ax + b < c or its complement |
| 1.5 | (-1.333, 5.333) | x = -1.333 and x = 5.333 | |ax + b| compared to c, solved from −c < ax + b < c or its complement |
| 2 | (-1.000, 4.000) | x = -1.000 and x = 4.000 | |ax + b| compared to c, solved from −c < ax + b < c or its complement |
| 3 | (-0.6667, 2.667) | x = -0.6667 and x = 2.667 | |ax + b| compared to c, solved from −c < ax + b < c or its complement |
| 4 | (-0.5000, 2.000) | x = -0.5000 and x = 2.000 | |ax + b| compared to c, solved from −c < ax + b < c or its complement |
| 6 | (-0.3333, 1.333) | x = -0.3333 and x = 1.333 | |ax + b| compared to c, solved from −c < ax + b < c or its complement |