Absolute Value Inequalities Calculator
Solve absolute value inequalities problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
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Adjust values & calculateFormula
For less-than: convert to compound inequality -c < ax + b < c (bounded interval). For greater-than: split into ax + b > c OR ax + b < -c (two unbounded rays). The coefficient a, constant b, and right-hand side c determine the boundary points and interval width.
Last reviewed: December 2025
Worked Examples
Example 1: Solving |3x - 6| < 9
Example 2: Solving |2x + 1| >= 7
Background & Theory
The Absolute Value Inequalities Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Absolute Value Inequalities Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
|ax + b| < c or |ax + b| > c
For less-than: convert to compound inequality -c < ax + b < c (bounded interval). For greater-than: split into ax + b > c OR ax + b < -c (two unbounded rays). The coefficient a, constant b, and right-hand side c determine the boundary points and interval width.
Worked Examples
Example 1: Solving |3x - 6| < 9
Problem: Find all values of x satisfying |3x - 6| < 9.
Solution: Convert to compound inequality: -9 < 3x - 6 < 9\nAdd 6 to all parts: -3 < 3x < 15\nDivide by 3: -1 < x < 5\nSolution in interval notation: (-1, 5)
Result: x is in the interval (-1, 5)
Example 2: Solving |2x + 1| >= 7
Problem: Find all values of x satisfying |2x + 1| >= 7.
Solution: Split into two cases:\nCase 1: 2x + 1 >= 7 => 2x >= 6 => x >= 3\nCase 2: 2x + 1 <= -7 => 2x <= -8 => x <= -4\nSolution: (-inf, -4] U [3, +inf)
Result: x <= -4 or x >= 3
Frequently Asked Questions
What is an absolute value inequality and how is it different from an equation?
An absolute value inequality involves an absolute value expression compared with a value using an inequality sign (such as less than or greater than), rather than an equals sign. While an absolute value equation like |ax + b| = c yields discrete point solutions, an inequality yields a continuous range or ranges of values. For less-than inequalities, the solution is a single bounded interval on the number line. For greater-than inequalities, the solution consists of two unbounded rays extending in opposite directions. Understanding this distinction is crucial for correctly interpreting and graphing solution sets in algebra.
How do you solve a less-than absolute value inequality?
To solve |ax + b| < c (where c is positive), you convert it into a compound inequality: -c < ax + b < c. This means the expression inside the absolute value must be between -c and c simultaneously. Then isolate x by subtracting b from all three parts and dividing by a (remembering to flip inequality signs if dividing by a negative number). The result is always a bounded interval. For example, |2x - 3| < 5 becomes -5 < 2x - 3 < 5, then -2 < 2x < 8, and finally -1 < x < 4, which is the open interval (-1, 4) on the number line.
What happens when an absolute value inequality has a negative right-hand side?
When the right-hand side is negative, the behavior depends on the inequality type. For |ax + b| < negative number, there is no solution because an absolute value is always non-negative, so it can never be less than a negative value. For |ax + b| > negative number, every real number is a solution because an absolute value is always greater than or equal to zero, which is always greater than any negative number. These special cases are important to recognize immediately because they require no algebraic manipulation and attempting to solve them normally can lead to incorrect or confusing results.
How do you express absolute value inequality solutions in interval notation?
Interval notation provides a compact way to describe solution sets. For bounded intervals from less-than inequalities, use parentheses for strict inequality and brackets for inclusive: (a, b) means a < x < b, while [a, b] means a <= x <= b. For unbounded intervals from greater-than inequalities, use the union symbol to join two rays: (-inf, a) union (b, +inf). Infinity always gets parentheses since it is not a number that can be reached or included. Mixed notation like (-inf, a] union [b, +inf) handles greater-than-or-equal-to cases. This notation is standard across mathematics and is essential for communicating solution sets precisely.
Can absolute value inequalities have no solution or all real numbers as solutions?
Yes, absolute value inequalities can produce these special solution sets beyond the standard interval results. The inequality |ax + b| < 0 has no solution since absolute values are always non-negative. Similarly, |ax + b| <= 0 has exactly one solution (when ax + b = 0). The inequality |ax + b| > 0 is satisfied by all real numbers except where ax + b = 0. The inequality |ax + b| >= 0 is satisfied by all real numbers without exception. Recognizing these cases quickly is a valuable algebra skill that saves significant time on homework, tests, and real-world applications.
How do you graph absolute value inequalities on a number line?
To graph absolute value inequality solutions on a number line, first solve the inequality to find the boundary points. For less-than inequalities yielding a bounded interval like -1 < x < 4, place open circles at -1 and 4 (or filled circles for less-than-or-equal) and shade the region between them. For greater-than inequalities yielding two rays like x < -1 or x > 4, place open circles at the boundary points and shade outward in both directions toward negative and positive infinity. The visual representation clearly shows whether the solution is a connected interval or two disconnected regions.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy