Absolute Value Inequality Calculator

Q: Solve |3x + 1| < 10.

−10 < 3x + 1 < 10 gives −11/3 < x < 3.

Q: Why do the two cases look so different?

'Close to a point' is one interval, while 'far from a point' is everything outside it — necessarily two pieces.

Work out absolute value inequality instantly with clear inputs, formula shown and shareable results.

About Absolute Value Inequality Calculator

An 'less than' absolute inequality gives a single bounded interval between the two critical values, while a 'greater than' one gives two rays heading outward from them.

Formulas

Less than

|u| < c ⇒ −c < u < c

Greater than

|u| > c ⇒ u < −c or u > c

Worked example

With the default values below, Absolute Value Inequality Calculator returns Solution of -3.666667 < x < 3. Change any input above and every figure here is recalculated in your browser.

Inputs used

Coefficient a in |ax + b| ? c: 3
Constant b: 1
Right-hand side c: 10
Relationone of |ax + b| < c, |ax + b| > c: |ax + b| < c

Results

Solution: -3.666667 < x < 3
Lower critical value: -3.66666667
Upper critical value: 3.00000000
Distance between the critical values: 6.66666667

Step by step

1Values used
2Less than
3Greater than
4Solution= -3.666667 < x < 3
5Lower critical value= -3.66666667
6Upper critical value= 3.00000000
7Distance between the critical values= 6.66666667

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Solve |3x + 1| < 10.