Algebra • Core Flagship Pillar

Inequality Calculator

Solve single-variable linear inequalities ($ax + b \le cx + d$), compound range inequalities ($L \le bx + c \le R$), and absolute value inequalities ($|ax + b| \le c$) with real-time number line graphing and interval notation.

Reviewed by Sanjay Samanta & Academic Review Board

Last Updated: September 2026

Verified Real Analysis Interval

Enter Inequality Expression

Preset Examples

Format: ax + b [Sign] cx + d

a (x)

Sign

c (x)

Real Number Line Solution Graph

Interval Notation

(-∞, 5]

Continuous Real Interval

Set Builder Notation

{ x | x ≤ 5 }

Algebraic Condition

∑

Step-by-Step Inequality Algebraic Derivation

Direct Answer & Overview

Verified Educational Guide

How to Solve an Algebraic Inequality

To solve a linear inequality ax + b ≤ cx + d, isolate variable terms on one side and constant terms on the other: (a − c)x ≤ (d − b). Divide by (a − c). If (a − c) is negative, reverse the inequality direction (e.g. ≤ becomes ≥). Express final solutions in interval notation [k, ∞) or (−∞, k] and plot on a real number line.

Primary Mathematical Formula Standard Mathematical Model

Standard Equation

ƒ(x)

Q.E.D.

ax + b \le cx + d \implies (a - c)x \le (d - b) \implies x \le \frac{d - b}{a - c} \quad (\text{flip if } a - c < 0)

Evaluated with exact mathematical formulation • Rigorously verified

Exact Formula

Input Parameters

Required

Coefficients a, b, c, d and chosen inequality relation (<, ≤, >, ≥)

Expression type (Linear, Compound 3-part, or Absolute Value)

Expected Outputs

Calculated

Exact boundary value, inequality solution, and interval notation

Set builder notation and interactive number line plot

Worked Numerical Example

Instant Verification

Solve -3x + 4 > 16

→ -3x > 12; divide by -3 (flip sign): x < -4

x < -4 (Interval: (-∞, -4))

Inequality Fundamentals & The Negative Sign Flip Rule

An algebraic inequality describes an order relationship between two expressions using the comparison symbols < (less than), ≤ (less than or equal), > (greater than), or ≥ (greater than or equal). Unlike equations with single numerical solutions, inequalities typically yield infinite sets of real numbers.

The Golden Rule of Inequalities

Whenever you multiply or divide both sides by a negative number, you MUST reverse (flip) the inequality symbol.

Proof: On the number line, 3 < 7. Multiplying by −1 reflects both values across 0 to −3 and −7. Since −3 is to the right of −7, −3 > −7.

Interval Notation vs. Set Builder Notation

Interval Notation

[a, b], (a, b), (-∞, k], [k, ∞)

Brackets [ ] indicate inclusive endpoints (solid dots on number line). Parentheses ( ) indicate strict exclusions and infinities (open circles).

Set Builder Notation

{ x ∈ ℝ | x ≥ k }

Reads: "The set of all real numbers x such that x is greater than or equal to k."

Compound Inequalities (AND vs. OR)

Conjunction (AND / Intersection)

a ≤ x ≤ b

Represents a bounded interval between two numbers. Both conditions must hold true simultaneously: x ≥ a AND x ≤ b.

Disjunction (OR / Union)

x < a or x > b

Represents two disconnected rays extending outward toward ±∞. Solution is the union: (-∞, a) ∪ (b, ∞).

Absolute Value Inequalities (|u| ≤ c vs. |u| ≥ c)

The absolute value $|u|$ measures distance from zero on the real number line:

"Less Than" Case (|u| ≤ c)

-c ≤ u ≤ c (Bounded Segment)

Distance from origin is within c units.

"Greater Than" Case (|u| ≥ c)

u ≤ -c or u ≥ c (Outer Rays)

Distance from origin is at least c units.

Real-World Applications of Inequalities

Engineering Tolerances & QA

Precision manufacturing uses absolute value inequalities $|x - \mu| \le \delta$ to specify acceptable part dimensions before rejection.

Budget Constraints & Finance

Financial planners set up budget inequalities (Fixed + Variable Expenses ≤ Income) to establish maximum discretionary spending limits.

Speed & Safety Limits

Traffic regulations and industrial robotics enforce compound speed constraints (45 ≤ Speed ≤ 65 mph) for safe operational flow.

Step-by-Step Worked Numerical Solutions

Example 1: Solving with Sign Reversal Negative Coeff

Problem: Solve 5 − 2x ≤ 13.

1. Subtract 5 from both sides: −2x ≤ 8.

2. Divide by −2 and FLIP the inequality sign: x ≥ 8 / (−2) ==> x ≥ −4.

3. Interval notation: [−4, ∞).

Solution: x ≥ −4 or [−4, ∞)

Common Pitfalls & Mistakes to Avoid

Forgetting to Flip Inequality Sign

Failing to reverse the inequality direction when dividing by negative coefficients is the #1 student error in algebra.

Using Brackets on Infinity

Infinity (∞ and −∞) is an unbounded concept, not a real number. Never write [k, ∞]; always use parentheses [k, ∞).

Negative Absolute Value Contradictions

|x| < −3 has No Solution (absolute value is never negative). In contrast, |x| > −3 is true for All Real Numbers.

Fact-Checked & Verified • Computational Accuracy Standards

Updated August 2026 • Editorial Policy

Authored By

Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By

Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

How do you solve a linear inequality step-by-step?

Solving a linear inequality follows the same principles as solving linear equations, with one crucial rule: whenever you multiply or divide both sides by a negative number, you must flip the inequality sign (e.g., < becomes >, and ≤ becomes ≥).

What is the difference between Interval Notation and Set Builder Notation?

Interval notation expresses solutions as continuous ranges using brackets [ ] for inclusive endpoints (≤, ≥) and parentheses ( ) for non-inclusive endpoints (<, >) and infinities (e.g., [2, 5) or (-∞, 4]). Set builder notation specifies the algebraic condition enclosed in braces, such as { x ∈ ℝ | x ≥ 3 }.

How do you solve compound inequalities (AND vs. OR)?

Compound inequalities joined by "AND" represent intersections where x must satisfy both conditions simultaneously (forming a bounded interval like [a, b]). Compound inequalities joined by "OR" represent unions where x satisfies either condition (forming two disjoint rays like (-∞, a] ∪ [b, ∞)).

How do you solve an absolute value inequality |u| ≤ c vs. |u| ≥ c?

For a positive constant c: |u| ≤ c represents a distance within c of zero, converted into a bounded compound inequality: −c ≤ u ≤ c. In contrast, |u| ≥ c represents a distance beyond c, converted into two disjoint inequalities: u ≤ −c OR u ≥ c.

Why does multiplying or dividing by a negative number flip the inequality symbol?

Multiplying by a negative number reverses order on the real number line. For example, since 2 < 5, multiplying both sides by −1 yields −2 > −5 (because −2 is farther to the right on the number line than −5).

How are inequalities used in real-world business budgets and engineering tolerances?

Engineers use inequalities to define allowable mechanical tolerances (e.g., |diameter − 50mm| ≤ 0.05mm). Financial managers use linear inequalities in linear programming to maximize profit subject to budget and labor constraints (e.g., Cost ≤ Budget).