Intervals and Inequalities Preview

A mobile plan charges Rs 299 per month for 2 GB of data, plus Rs 12 for every extra GB. You have a budget of Rs 500. How many extra gigabytes can you use? The constraint is 299 + 12g \le 500, and the answer is not a single number — it is every value of g from 0 up to 16.75. The answer to an inequality is a set of numbers, and that set is almost always an interval.

This article is the notation and the toolkit: the four types of interval, bracket notation, how to solve linear inequalities (including the critical trap when you multiply by a negative), and how to handle absolute-value inequalities — the type that produces a bounded interval instead of a ray.

What is an interval?

An interval is a subset of \mathbb{R} with no gaps. Formally: a set I of real numbers is an interval if, whenever a and b are in I with a < b, every number between a and b is also in I.

The set \{1, 2, 3\} is not an interval — it contains 1 and 3 but not 2.5. The set of all x with 1 \le x \le 3 is an interval — it contains every real number from 1 to 3 with no missing points.

The four bounded intervals

Given two real numbers a < b, there are four ways to build a bounded interval depending on which endpoints you include.

Notation	Name	Set	Endpoints
[a, b]	Closed	\{x : a \le x \le b\}	Both included
(a, b)	Open	\{x : a < x < b\}	Both excluded
[a, b)	Half-open (closed-open)	\{x : a \le x < b\}	Left included, right excluded
(a, b]	Half-open (open-closed)	\{x : a < x \le b\}	Left excluded, right included

The square bracket [ means "include this endpoint" and the round bracket ( means "exclude it." The same convention holds everywhere in mathematics, and it maps directly to the filled-dot / hollow-dot picture on the number line: a filled dot at the endpoint means it is in the set, a hollow dot means it is not.

Unbounded intervals and the infinity symbol

Not every interval has two finite endpoints. The solution to x < 5 is a ray extending to the left forever. The notation uses -\infty and +\infty (or just \infty) as stand-ins for "keeps going."

Notation	Set	Description
(-\infty, b)	\{x : x < b\}	Everything left of b
(-\infty, b]	\{x : x \le b\}	Everything left of and including b
(a, \infty)	\{x : x > a\}	Everything right of a
[a, \infty)	\{x : x \ge a\}	Everything right of and including a
(-\infty, \infty)	\mathbb{R}	The entire real line

An important convention: \infty always gets a round bracket, never a square one. You write (-\infty, 5], not [-\infty, 5]. The reason is that \infty is not a real number — it is a notational device meaning "no boundary here." Since \infty is not an element of \mathbb{R}, it cannot be "included" in the interval, so a square bracket would be meaningless.

Solving linear inequalities

The rules for manipulating inequalities are almost the same as for equations. You can add or subtract the same number on both sides. You can multiply or divide both sides by the same positive number. The one critical difference — drilled into you in Operations and Properties — is that multiplying or dividing by a negative number reverses the inequality sign.

Here is the complete set of legal moves:

• If a < b, then a + c < b + c for any c.
• If a < b and c > 0, then ac < bc.
• If a < b and c < 0, then ac > bc — the sign flips.

The same rules hold for \le, >, and \ge.

A simple example without the flip

Solve 3x + 7 \le 22.

Subtract 7: 3x \le 15. Divide by 3 (positive, no flip): x \le 5. The solution is (-\infty, 5].

An example with the flip

Solve -4x + 1 > 9.

Subtract 1: -4x > 8. Divide by -4 (negative — flip): x < -2. The solution is (-\infty, -2).

If you had forgotten to flip, you would have written x > -2, which is the interval (-2, \infty) — the entire wrong half of the number line. This is the most common error in inequality problems, and exam graders in board exams and JEE see it every year.

Absolute value inequalities

The absolute value |x| is the distance of x from 0 on the number line. More generally, |x - c| is the distance of x from c. This geometric reading is the key to solving absolute-value inequalities.

The "less than" case: |x - c| < r

The inequality |x - c| < r says "the distance from x to c is less than r." Geometrically, that means x lies within r units of c — which is the open interval (c - r,\ c + r).

The "greater than" case: |x - c| > r

The inequality |x - c| > r says "the distance from x to c is more than r." The point x must be outside the interval of radius r around c — so the solution is two rays:

In interval notation: (-\infty,\ c - r) \cup (c + r,\ \infty).

Combining intervals: intersection and union

Sometimes solving an inequality produces two conditions that must both hold (intersection, \cap) or at least one must hold (union, \cup).

• Intersection A \cap B: the set of numbers in both A and B. Geometrically, this is the overlap.
• Union A \cup B: the set of numbers in A or B (or both). Geometrically, this is the combined shading.

For example, [1, 5] \cap [3, 8] = [3, 5] — the overlap of the two intervals. And (-\infty, -2) \cup (2, \infty) is the solution to |x| > 2: two rays with a gap in the middle.

The intersection of two intervals is always an interval (or empty). The union of two intervals is an interval only when they overlap or share an endpoint; otherwise it is two disjoint pieces and you write it with \cup.

An interactive exploration

Drag the red point along the number line below. The readout shows x, |x - 3|, and whether the point satisfies the inequality |x - 3| \le 4 — that is, whether x lies in the interval [-1, 7].

Two worked examples

Common confusions

• "(3, 7) is a point in the plane." In the context of intervals, (3, 7) means the open interval from 3 to 7. In coordinate geometry, (3, 7) means the point with x-coordinate 3 and y-coordinate 7. The notation is identical; the meaning depends on context. When there is any risk of confusion, you can write ]3, 7[ (the French convention for open intervals) or explicitly say "the interval (3, 7)."

• "\infty is a number, so [\infty) makes sense." No. \infty is not a real number — it is a shorthand meaning "no upper boundary." You cannot include something that does not exist in a set, so the bracket next to \infty is always round. Writing [3, \infty] is a notational error.

• "When I multiply an inequality by a negative number, only one side flips." Both sides change, and the inequality symbol itself flips. If a < b and you multiply by -1, you get -a > -b. The direction of the arrow reverses. Forgetting the flip is the single most common mistake in inequality problems.

• "|x| < -2 has all real numbers as solutions." It has no solutions. The absolute value is always non-negative, so |x| \ge 0 > -2 for every x, making |x| < -2 impossible. Conversely, |x| > -2 is true for every x, so the solution set is \mathbb{R}.

• "To solve |x - 3| \le 5, just drop the absolute value and solve x - 3 \le 5." That gives only the right half of the answer (x \le 8). You also need the condition x - 3 \ge -5, i.e., x \ge -2. The full answer is -2 \le x \le 8, the interval [-2, 8]. Dropping the absolute value without considering both cases always loses half the solution.

• "Open and closed intervals contain the same numbers." Almost, but not quite. (0, 1) and [0, 1] differ by exactly two elements — the endpoints 0 and 1. For most purposes this difference is negligible, but it matters when you ask "does the function achieve its maximum on this interval?" On [0, 1] it always does (by the extreme value theorem). On (0, 1) it might not, because the maximum might occur at an excluded endpoint.

Going deeper

Intervals and completeness

The fact that intervals have no internal gaps is a direct consequence of the completeness of \mathbb{R}. In \mathbb{Q}, the set \{q \in \mathbb{Q} : 1 < q < 2\} looks like an interval but has holes — \sqrt{2} is missing, for instance. When you move to \mathbb{R}, those holes fill in, and every interval is genuinely gap-free. The definition of an interval — "if a and b are in I, so is everything between them" — only gives you a nice connected stretch because \mathbb{R} is complete. On the rationals, the "everything between them" part fails for irrational points.

Compound inequalities and systems

A single inequality gives a single interval or ray. A system of inequalities — two or more inequalities that must hold simultaneously — gives an intersection of intervals. For example, the system

gives x \in (1, 5]. The solution is the overlap of (1, \infty) and (-\infty, 5].

When the two conditions point in opposite directions — like x > 7 and x < 3 — the intersection is empty, and the system has no solution. Geometrically, the two rays don't overlap.

When the conditions are joined by "or" instead of "and," you take the union. x < -1 or x > 4 gives (-\infty, -1) \cup (4, \infty).

Quadratic and polynomial inequalities

Once you move beyond linear expressions, the solution sets can be unions of intervals. The inequality x^2 - 4 > 0 factors as (x - 2)(x + 2) > 0, which holds when both factors are positive or both are negative. The solution is (-\infty, -2) \cup (2, \infty). The technique — factor, find the roots, test the sign in each interval — is the subject of Quadratic Inequalities, and it builds directly on the interval language from this article.

The triangle inequality revisited

The absolute-value inequality |a + b| \le |a| + |b| from Operations and Properties is the reason the "distance" interpretation of |x - c| works. That inequality says that distances on the number line obey the triangle inequality, which is the minimum requirement for something to be a genuine distance function (a metric). Intervals, as subsets of \mathbb{R} defined by distance conditions like |x - c| < r, inherit their clean geometric behaviour from this single inequality.

Where this leads next

Intervals and inequalities are the language in which most of the rest of algebra states its answers.

• Real Numbers — Properties — the completeness and density results that make intervals gap-free.
• Operations and Properties — the inequality rules (especially the sign-flip) and the triangle inequality that power absolute-value problems.
• Sets — Introduction — the formal set-builder notation \{x : \ldots\} that interval notation abbreviates.
• Algebraic Expressions — the next chapter, where variables and operations combine into expressions that inequalities will eventually constrain.
• Quadratic Inequalities — where the solution sets become unions of intervals, and sign-charts replace single-step algebra.