About this absolute value calculator
This absolute value calculator has two modes: evaluate the absolute value of one real number, or solve an equation or inequality containing the absolute value of a linear expression. The solver uses the supported form k|ax + b| + d compared with c, including an outside multiplier and constant.
The answer includes exact solution points or interval notation, with steps showing isolation, the appropriate branch rule and boundary checks. This is different from the existing Inequality Calculator, which handles linear and quadratic comparisons without this absolute-value form. This page does not accept arbitrary symbolic expressions.
How to solve an absolute value equation or inequality
- Choose Solve equation / inequality, or Evaluate one number.
- For solving, enter the five coefficients. The initial k = 1 and d = 0 describe the simpler form |ax + b| compared with c.
- Select =, <, ≤, > or ≥. Enter zero explicitly for any missing coefficient.
- Calculate and read both the solution notation and the working. A negative outside multiplier reverses an inequality when it is divided out.
In Evaluate mode, enter one supported decimal or fraction instead. Changing any input clears the previous result; switching modes hides and disables unused fields. Reset restores the initial solver form. Copy, Download and Print retain the original comparison and calculation steps.
For an ordinary equation without absolute value, use the Linear Equation Calculator.
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For a separate arithmetic check, you may browse scientific calculators (paid link). Check the model’s supported functions; ordinary scientific models do not necessarily solve symbolic absolute-value equations or perform polynomial long division. Check supported functions and use this page’s exact working for the method. No purchase is required.
Isolation, equation branches and interval rules
The solver first subtracts d and divides by a nonzero k. A negative k reverses < and >, or ≤ and ≥; equality keeps the same relation. It then examines the isolated bound before splitting into linear branches. Constant cases are evaluated directly instead of dividing by zero.
For a nonnegative bound t, |ax + b| = t gives two possible linear equalities. A less-than comparison selects the inside interval; a greater-than comparison selects the outside regions joined by OR. A zero bound can collapse solutions to one point or exclude that point. A negative bound must be handled separately because an absolute value cannot be negative.
The branch boundaries are kept as reduced fractions. The existing Fraction Calculator can help you check individual arithmetic operations, but it does not determine the absolute-value solution set.
Worked absolute value examples
To solve |2x − 3| = 7, enter k = 1, a = 2, b = −3, d = 0 and c = 7. The branches give x = −2 or x = 5, so the answer is {−2, 5}. Changing only the comparison to ≤ gives the interval [−2, 5].
For −2|3x + 1| + 4 ≥ −10, enter k = −2, a = 3, b = 1, d = 4 and c = −10. Subtracting 4 and dividing by −2 reverses the comparison, producing |3x + 1| ≤ 7. The solution is [−8/3, 2].
Evaluate −5/3 in the numeric mode and the exact result is 5/3. For |x| < 0, there is no real solution; for |x| ≥ 0, every real number is a solution. Use the Scientific Calculator for a separate numerical substitution check if helpful.
Reading and verifying the result
Braces list equality solutions, such as {−2, 5}. Square brackets include a finite boundary and parentheses exclude it. The symbol ∪ joins separate regions, ∅ means no real solution, and (−∞, ∞) includes every real number. Infinity is never an included endpoint.
The solver substitutes each finite equality boundary into the original outside-multiplier expression, not merely its simplified branch. This checks the values exactly. An inequality’s endpoint inclusion still depends on whether its selected comparison is strict or inclusive. You can also substitute an interior point and an exterior point to practise interpreting the interval.
Supported forms and input limits
The solver accepts one absolute value containing ax + b, an outside numeric multiplier k, an outside constant d and a numeric right side c. It does not solve two absolute-value terms, nested absolute values, nonlinear contents, variable right sides or general compound expressions. Rewrite a problem only when the rewritten form is mathematically equivalent.
Each input is limited to 30 characters. Integers, decimals with up to six places and simple fractions are accepted. Magnitudes must be within −10,000 to 10,000; fraction numerators and denominators must also be within that range, with a nonzero denominator. Commas, scientific notation and full expressions are not accepted in coefficient fields.
All accepted inputs are represented exactly as rational numbers. Results can therefore be fractions even when some inputs were decimals. Numeric absolute value means a real number’s distance from zero; complex-number magnitude is a different operation and is outside this page’s scope. This educational solver does not infer units or decide which comparison models a real-world situation.
Absolute Value Calculator FAQs
Sources and methodology
These educational references explain the mathematical ideas. This independently implemented calculator is not affiliated with their publisher. Its supported methods, exact checks and limits are documented above.