Algebraic Solutions
A linear inequality in one variable is a mathematical statement that compares a linear expression to a value using one of the symbols <, >, ≤, or ≥. Unlike a linear equation, which has at most one solution, a linear inequality typically has infinitely many solutions that form an interval on the number line. For example, the inequality 2x + 3 > 7 is satisfied by every real number greater than 2, not just a single value. Solving a linear inequality algebraically means applying a sequence of reversible operations to isolate the variable, much like solving a linear equation. The critical difference is the multiplication (or division) rule: whenever you multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed. This rule is the source of most errors students make and deserves special attention. The solution set of a linear inequality can be expressed in three equivalent ways: set-builder notation such as {x ∈ ℝ | x > 2}, interval notation such as (2, ∞), or graphically as a ray on the number line. All three representations communicate the same information, and fluency with each is essential for higher mathematics including calculus, optimisation, and real analysis.
- Simplify Each Side
- Collect Variable Terms on One Side
- Isolate the Variable Term
- Divide (or Multiply) to Solve — Watch the Sign!
- Write and Graph the Solution Set
Simplify Each Side
Distribute any constants, combine like terms, and clear fractions or decimals if present. For instance, in 3(x − 2) + 4 < 10 − x, first distribute to get 3x − 6 + 4 < 10 − x, then combine to get 3x − 2 < 10 − x.
Collect Variable Terms on One Side
Add or subtract variable terms so they appear on one side only. Using the example: add x to both sides to get 4x − 2 < 10. Adding or subtracting never changes the direction of the inequality.
Isolate the Variable Term
Add or subtract constants to leave the variable term alone. Here, add 2 to both sides: 4x < 12. The inequality direction is unchanged.
Divide (or Multiply) to Solve — Watch the Sign!
Divide both sides by the coefficient of x. If the coefficient is positive, the direction stays the same: 4x < 12 gives x < 3. If the coefficient were −4 (e.g., −4x < 12), dividing by −4 reverses the inequality: x > −3. This reversal rule is the most important rule in working with inequalities.
Write and Graph the Solution Set
Express the answer in interval notation and/or set-builder notation. For x < 3, write (−∞, 3) or {x | x < 3}. On a number line, draw an open circle at 3 (since 3 is not included) and shade the ray extending to the left. For ≤ or ≥, use a closed (filled) circle.
The Key Equation
- a — Coefficient of x (dimensionless): The real-number multiplier of the variable. Its sign determines whether the inequality flips when you divide.
- b — Constant term on left side (dimensionless): A real number added to the variable term. It is eliminated by subtracting b from both sides.
- c — Constant on right side (dimensionless): The real number the expression is compared to. After subtracting b, it becomes c − b.
- \lessgtr — Inequality symbol (—): Any of <, >, ≤, ≥. The symbol may reverse when multiplying or dividing by a negative number.
Worked Example
Problem. Solve for x and express the solution in interval notation: −3x + 5 ≥ 14
Answer. x ≤ −3, written in interval notation as (−∞, −3]
Origins · 1 minThomas Harriot and how this was discovered
Thomas Harriot was an English mathematician and astronomer who introduced the modern inequality symbols < and > in his posthumously published work Artis Analyticae Praxis (1631). Before Harriot's notation, mathematicians described inequalities entirely in words or with cumbersome verbal phrases. His compact symbols made algebraic manipulation of inequalities far more efficient and are still used unchanged today. Harriot made telescopic observations of the Moon about the same time as Galileo and produced some of the earliest known telescopic maps of the lunar surface, yet he never published them during his lifetime.
Related Concepts
- Linear Equations in One Variable — Solving linear inequalities uses nearly identical algebraic steps to solving linear equations; understanding equations first builds the foundation for inequalities.
- Compound Inequalities — Compound inequalities combine two linear inequalities with 'and' or 'or', extending the single-variable techniques learned here.
- Interval Notation — Interval notation is the standard way to record the infinite solution sets produced by linear inequalities.
- Absolute Value Inequalities — Absolute value inequalities are solved by splitting them into two linear inequalities, directly applying the skills practiced here.