Linear inequalities in one unknown

A linear inequality in one unknown compares a linear expression with a value or another linear expression using symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). Instead of finding one exact value, we usually find a set of values that make the statement true.

Inequality symbols

Solving linear inequalities

Solve them much like linear equations: expand brackets when needed, collect like terms, then isolate the variable.

Example. Solve \(3x - 5 \leq 16\).

Add \(5\) to both sides: \(3x \leq 21\). Divide by \(3\): \(x \leq 7\).

The sign-flip rule

When you multiply or divide both sides of an inequality by a negative number, the inequality sign must reverse.

Example. Solve \(-2x + 1 > 9\).

Subtract \(1\): \(-2x > 8\). Divide by \(-2\), so reverse \(>\) to \(<\): \(x < -4\).

Number line and interval notation

The solution set can be written on a number line or in interval notation:

Compound inequalities

Compound inequalities combine two inequalities with and or or.

Example (and). Solve \(2 < x + 1 \leq 6\).

Subtract \(1\) from all three parts: \(1 < x \leq 5\). Interval form: \((1, 5]\).

Example (or). Solve \(x - 4 < -7\) or \(3x + 1 \geq 10\).

First: \(x < -3\). Second: \(3x \geq 9\), so \(x \geq 3\). Final set: \((-\infty, -3) \cup [3, \infty)\).

Word problems

Translate the condition into an inequality, solve, and interpret the answer in context.

Example. A bus ticket costs \$12. You have at most \$90. How many tickets \(n\) can you buy?

\(12n \leq 90 \Rightarrow n \leq 7.5\). Since \(n\) is a whole number of tickets, the maximum is \(7\).

History

Ideas behind inequalities are ancient. Greek mathematicians such as Euclid compared magnitudes and proved strict order relationships long before modern algebraic symbols. In India and the Islamic world, later algebra texts worked with bounds and constraints in practical problems involving trade, measurement, and inheritance.

The modern symbols \(<\) and \(>\) were introduced in the 17th century by Thomas Harriot. As algebra notation became standard, inequalities became a compact way to describe ranges of possible values rather than one exact answer. Today, the same language appears in budgeting, engineering safety margins, and optimization.

Derivation

Why does the sign flip with a negative?

Suppose \(a < b\). Subtract \(b\) from both sides to get \(a-b < 0\). Multiplying by \(-1\) gives \(b-a > 0\), so \(-a > -b\). Geometrically, multiplying by \(-1\) reflects numbers across zero on the number line, reversing left-right order. That is why dividing by a negative must reverse \(<\) to \(>\) (and \(\leq\) to \(\geq\)).

Why intervals match inequality endpoints

A strict inequality such as \(x < c\) excludes \(c\), so interval notation uses an open bracket \((\cdot)\). A non-strict inequality such as \(x \leq c\) includes \(c\), so interval notation uses a closed bracket \([\cdot]\). This bracket choice encodes exactly the same endpoint logic as open and closed dots on number lines.

Checkpoints

Pause and reason before moving on. Discuss with a classmate or write your reasoning in a notebook.

Applications

Budgeting and spending limits

If each notebook costs \$3 and you have at most \$24, the purchase count \(n\) satisfies \(3n \leq 24\). Inequalities model spending caps in daily life, event planning, and inventory decisions.

Safety and engineering limits

Many systems run within allowable ranges, such as pressure \(P \leq P_{\max}\) or temperature \(T_{\min} \leq T \leq T_{\max}\). Inequalities represent these safe operating regions directly.

Grades and performance thresholds

A pass condition might be "score at least 50," written as \(s \geq 50\). Scholarship criteria and qualification cutoffs are also inequality-based rules.

References

• Harriot, T. (1631). Artis analyticae praxis (early use of \(<\) and \(>\)).
• Euclid, Elements, Book V (order and comparison of magnitudes).
• Secondary school algebra curricula on linear inequalities and number-line representation.