Inequality Properties & Comparison
An inequality is a mathematical statement that compares two expressions using one of the symbols <, >, ≤, or ≥. Unlike equations, which assert that two quantities are equal, inequalities describe a range of possible values. For example, the statement x > 3 means that x can be 3.1, 4, 100, or any number strictly greater than 3 — an infinite collection of solutions rather than a single answer. Understanding how to work with inequalities is essential throughout algebra, calculus, and real-world applications such as budgeting, engineering tolerances, and optimization problems. The properties of inequalities govern how you can legally manipulate them while keeping the comparison true. Most of these properties mirror those of equations — you can add or subtract the same value from both sides, or multiply and divide by a positive number — but one critical rule has no counterpart in equations: multiplying or dividing both sides by a negative number reverses the direction of the inequality symbol. Forgetting this rule is the single most common source of errors when solving inequalities. Comparing values is the foundation of ordering the real number line. Techniques include using the number line itself, computing differences (a − b and checking its sign), and applying transitivity (if a < b and b < c, then a < c). This article builds a complete toolkit of inequality properties and comparison strategies that will support every topic that follows in algebra and beyond.
- Addition and Subtraction Property
- Multiplication and Division by a Positive Number
- Multiplication and Division by a Negative Number — Flip the Symbol
- Transitive Property
- Comparison by Difference
- Non-Strict vs. Strict Inequalities and Boundary Values
Addition and Subtraction Property
If a < b, then a + c < b + c for any real number c, and a − c < b − c. Adding or subtracting the same quantity from both sides preserves the inequality. Example: since 3 < 7, adding 4 to both sides gives 7 < 11, which is still true. This property is used to isolate variables by moving constants across the inequality sign — exactly like solving equations.
Multiplication and Division by a Positive Number
If a < b and c > 0, then ac < bc and a/c < b/c. Multiplying or dividing both sides by a positive number preserves the direction of the inequality. Example: since 2 < 6, multiplying by 3 gives 6 < 18 (still true). You can use this property freely when the multiplier or divisor is confirmed to be positive.
Multiplication and Division by a Negative Number — Flip the Symbol
If a < b and c < 0, then ac > bc and a/c > b/c. Multiplying or dividing both sides by a negative number reverses the inequality. Example: 2 < 6, but multiplying by −1 gives −2 > −6 (reversed). This happens because negative numbers reverse the order of the number line. Always flip the inequality symbol when you multiply or divide by a negative value.
Transitive Property
If a < b and b < c, then a < c. This property allows chains of comparisons. For example, if you know that x < 5 and 5 < 12, you can immediately conclude x < 12 without extra work. Transitivity is what makes the real number line a total order — every pair of real numbers can be compared, and those comparisons chain consistently.
Comparison by Difference
A reliable technique for comparing two numbers a and b is to compute a − b. If a − b > 0 then a > b; if a − b < 0 then a < b; if a − b = 0 then a = b. This technique is especially useful when comparing fractions or algebraic expressions. For instance, to compare 7/9 and 5/7, compute 7/9 − 5/7 = 49/63 − 45/63 = 4/63 > 0, so 7/9 > 5/7.
Non-Strict vs. Strict Inequalities and Boundary Values
A strict inequality (< or >) excludes the boundary value, while a non-strict inequality (≤ or ≥) includes it. On a number line, strict inequalities are shown with an open circle and non-strict with a closed (filled) circle. When solving x − 4 ≤ 2, the solution is x ≤ 6, and x = 6 is included. When solving x − 4 < 2, the solution is x < 6, and x = 6 is excluded. This distinction matters enormously in problems about feasible regions and interval notation.
The Key Equation
- a — First value (dimensionless): Any real number being compared
- b — Second value (dimensionless): Any real number being compared
- a - b — Difference (dimensionless): The signed difference whose sign determines the order relationship between a and b
Worked Example
Problem. Solve the inequality −3x + 5 > 14 and graph the solution on a number line.
Answer. x < −3, or equivalently (−∞, −3)
Origins · 1 minThomas Harriot and how this was discovered
Thomas Harriot is credited with introducing the modern inequality symbols < and > in his posthumously published work Artis Analyticae Praxis (1631). Before Harriot, mathematicians used cumbersome verbal descriptions or other notations to express order relationships. His compact symbols became universally adopted and are the same ones students use today, making inequality statements far more efficient to write and read. Harriot was also one of the first people to use a telescope to observe the Moon — even before Galileo published his lunar drawings — but he never published most of his scientific work during his lifetime.
Related Concepts
- Solving Linear Inequalities — Directly applies the inequality properties covered here to find solution sets for one-variable linear inequalities.
- Compound Inequalities — Combines two inequalities using 'and' or 'or', requiring mastery of individual inequality properties and comparison techniques.
- Absolute Value Inequalities — Uses inequality comparison and the sign-flip property when splitting absolute value statements into two separate inequalities.
- The Real Number Line — The geometric foundation for understanding order, comparison, and the direction of inequalities among real numbers.
- Interval Notation — The standard shorthand for expressing solution sets of inequalities, distinguishing open and closed endpoints.