Linear Inequality
A linear inequality is a mathematical statement that compares two linear expressions using an inequality symbol such as <, >, ≤, or ≥. Unlike a linear equation, it defines a range of values that satisfy the condition rather than a single solution. Linear inequalities are widely used in optimisation problems, budgeting, and real-world constraints.
Key Formula
ax + b < c (or >, ≤, ≥)
LaTeX: ax + b < c \quad \text{(or } >, \leq, \geq\text{)}
| Symbol | Meaning | Unit |
|---|---|---|
| a | Coefficient of the variable | dimensionless |
| x | Unknown variable | dimensionless |
| b | Constant term | dimensionless |
| c | Right-hand side constant | dimensionless |
Worked Example
Problem
Solve the inequality 3x + 5 ≤ 20 and represent the solution on a number line.
Solution
Step 1: Subtract 5 from both sides: 3x ≤ 15. Step 2: Divide both sides by 3: x ≤ 5. Step 3: The solution set is all real numbers less than or equal to 5.
Answer
x ≤ 5, represented as (-∞, 5] on the number line.
Inequality Symbols and Their Meanings
| Symbol | Meaning | Example | Solution Type |
|---|---|---|---|
| < | Strictly less than | x < 3 | Open boundary |
| > | Strictly greater than | x > 3 | Open boundary |
| ≤ | Less than or equal to | x ≤ 3 | Closed boundary |
| ≥ | Greater than or equal to | x ≥ 3 | Closed boundary |
| ≠ | Not equal to | x ≠ 3 | Excluded point |
Interactive Tools
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Related Terms
Linear Equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the first power, producing a straight-line graph when plotted. The standard form of a linear equation in one variable is ax + b = 0, while in two variables it is ax + by = c. Linear equations are foundational in algebra and appear throughout science, economics, and engineering for modelling proportional relationships.
Absolute Value
The absolute value of a real number is its distance from zero on the number line, always expressed as a non-negative quantity. It strips the sign from a number, so |x| = x if x ≥ 0 and |x| = −x if x < 0. Absolute value is fundamental in measuring distances, errors, and deviations in mathematics, physics, and engineering.
From Latin "linearis" (belonging to a line) and "inaequalitas" (unevenness or disparity). The term entered formal mathematical usage in the 18th century as algebra formalised the study of inequalities alongside equations.