Union and Intersection of Intervals

Interval notation is the standard shorthand for describing sets of real numbers, and the union and intersection of intervals are the two operations you will use to combine those sets. When you solve an inequality or find a function's domain, the answer is rarely a single neat interval. More often, you get two or more separate pieces, and the union symbol ∪ or the intersection symbol ∩ tells you how those pieces fit together. The union means "or": a number belongs to the combined set if it is in either interval. The intersection means "and": a number must be in both intervals. Getting this distinction right is the difference between a correct answer and a subtle, grade-costing error. You can combine intervals correctly using number-line diagrams and worked examples.

The union of intervals, written with ∪, collects every number that lies in at least one of the intervals. Think of it as the word "or". If the solution to an inequality is x is less than 2 or x is greater than or equal to 5, then in interval notation you write (−∞, 2) ∪ [5, ∞). The two intervals do not touch, and the union symbol is what holds them together as one answer set. Without the ∪, writing (−∞, 2) [5, ∞) is meaningless; the notation simply does not parse. The union is not optional. It is the grammatical glue that makes separate intervals into a single, valid expression of a solution set.

When the intervals overlap, the union collapses into a single interval. For example, [1, 5] ∪ [3, 7] is just [1, 7], because every number from 3 to 5 is already covered, and the union adds nothing new beyond what the larger span already includes. Many students write the longer form out of caution, but the simplified version is the answer a grader or a calculator expects. The rule is simple: if the intervals share any point, merge them into one continuous interval from the smaller left endpoint to the larger right endpoint. If they do not share a point, keep the ∪ and list the intervals in ascending order of their left endpoints.

The intersection of intervals, written with ∩, keeps only the numbers that belong to both intervals at once. Think of it as the word "and". For example, [1, 5] ∩ (3, 7) = (3, 5]. The overlapping region runs from just above 3 to exactly 5. The left endpoint is open because 3 is not in the first interval, and the right endpoint is closed because 5 is in both. When the intervals do not overlap at all, the intersection is the empty set, written ∅. So [1, 2] ∩ [3, 4] = ∅. No real number is simultaneously in both, so the answer is no numbers at all.

Intersection rarely appears in basic inequality solving, because a single inequality like x > 2 and x < 5 is usually written as the compound form 2 < x < 5, which is a single interval (2, 5). But intersection becomes essential when you work with absolute value inequalities, where the solution to |x| < a is the intersection of two inequalities, −a < x < a. OpenStax College Algebra 2e, section 2.7, covers this conversion. The key habit to build is checking whether the word "and" or "or" governs the original problem. An "and" inequality always produces an intersection, which may be a single interval, a smaller interval, or the empty set.

Whether you union or intersect, the first step is always to determine if the intervals overlap. Draw them on a number line. Shade the first interval in one direction, the second in another. The union is everything shaded in either direction. The intersection is only where both shades overlap. If the shades never meet, the intersection is ∅. This visual check catches the most common error in interval work: writing a union when the intervals actually touch. For instance, [1, 5] and (5, 7] share no point, because 5 is included in the first but excluded from the second. The union is [1, 5] ∪ (5, 7], and the intersection is ∅.

Another frequent mistake is assuming that touching endpoints mean the intervals overlap. The intervals [1, 5] and [5, 7] do overlap, because 5 is in both. Their union is [1, 7], and their intersection is {5}, a single point. But {5} is not an interval; it is a set with one element. Interval notation cannot represent a single point, so you would write the intersection as {5} or, if the context demands an interval, you would note that no interval exists. This distinction matters in calculus, where intervals of increase or decrease are almost always open, and a single point of overlap can change whether a function is continuous on a closed interval.

The complement of an interval is everything not in it, relative to the real numbers. The complement of (a, b) is (−∞, a] ∪ [b, ∞). Notice the parentheses flip to brackets, because the endpoints themselves move out of the interval. The complement of [a, b] is (−∞, a) ∪ (b, ∞). This operation is a union of two disjoint intervals, and it is where the union symbol earns its keep. Difference, written as A minus B, is the set of numbers in A but not in B. It is not a separate operation; you compute it as the intersection of A with the complement of B. For example, [1, 10] minus (3, 5) = [1, 3] ∪ [5, 10].

The complement is essential for writing the domain of functions that exclude points. The domain of f(x) = 1/(x² − 4) is all real numbers except −2 and 2. In interval notation, that is (−∞, −2) ∪ (−2, 2) ∪ (2, ∞). Most students write the first and third intervals correctly but forget the middle one, producing (−∞, −2) ∪ (2, ∞), which wrongly includes numbers between −2 and 2. The complement of the set {−2, 2} is not a single interval; it is three. Drawing the number line and shading everything except the two points makes the three-interval structure obvious.

Difference also shows up in piecewise-defined domains. If a function is defined on [0, 10] except at x = 4, the domain is [0, 4) ∪ (4, 10]. This is a difference operation: the interval [0, 10] minus the point {4}. The result is two intervals, not one, because the single missing point splits the interval. The union symbol is mandatory here. Writing [0, 4) (4, 10] is not a valid expression. The reader or the calculator will reject it. Always check whether the result of a difference has more than one piece, and if so, join the pieces with ∪.

Interval notation union appears in every domain problem that involves division by zero or an even root. The domain of f(x) = √(x − 1) is [1, ∞), a single interval. The domain of g(x) = √(x − 1)/(x − 3) is [1, 3) ∪ (3, ∞), because x = 3 makes the denominator zero. The union is necessary because the function is defined on two separate sides of the gap. OpenStax College Algebra 2e, section 1.1, establishes the notation: brackets for included endpoints, parentheses for excluded, and ∞ always with a parenthesis because infinity is not a number.

When you solve a compound inequality with "or", the solution is a union. For example, x < −1 or x ≥ 3 becomes (−∞, −1) ∪ [3, ∞). The word "or" is the signal. When the inequality uses "and", as in x > −1 and x < 3, the solution is the single interval (−1, 3). This mapping from words to symbols is the core skill. The phrase "compound inequalities interval notation" is just a label for this conversion; the actual method is to translate each inequality separately, then combine with ∪ for "or" or ∩ for "and". Practice with a number line until the operation feels automatic.

One warning: do not use ∪ when the intervals overlap, because the union then simplifies to a single interval, and writing the longer form suggests you did not notice the overlap. The complement of an interval, likewise, is best found by drawing the line first. If you try to do it algebraically, you will likely miss a bracket-to-parenthesis flip. For instance, the complement of (−∞, 2] is (2, ∞), not [2, ∞). The endpoint 2 was included in the original, so it must be excluded from the complement. A number line shows this instantly.

Absolute value inequalities are the most common source of intersections in a typical course. The inequality |x| < 3 means x is between −3 and 3, written as (−3, 3). This is the intersection of x > −3 and x < 3. The inequality |x| ≥ 3 means x ≤ −3 or x ≥ 3, written as (−∞, −3] ∪ [3, ∞). OpenStax College Algebra 2e, section 2.7, shows this conversion explicitly. The critical habit is to write the two separate inequalities first, then decide whether to intersect or union them based on the original problem's wording.

Failure to distinguish these two cases produces the most common error in interval notation: writing ∪ when ∩ is meant, or vice versa. A good check is to test a number. For |x| < 3, test x = 0. It works, so 0 must be in the interval. For |x| ≥ 3, test x = 0. It fails, so 0 must be excluded. If your written interval includes 0 in the second case, you have used the wrong operation. This test takes five seconds and catches most mistakes.

When the absolute value inequality has a variable expression inside, solve it as a compound inequality. For |2x − 1| ≤ 5, write −5 ≤ 2x − 1 ≤ 5, then solve to get −2 ≤ x ≤ 3, which is [−2, 3]. This is an intersection of two conditions, but the result is a single interval. For |2x − 1| > 5, write 2x − 1 < −5 or 2x − 1 > 5, solve to get x < −2 or x > 3, which is (−∞, −2) ∪ (3, ∞). The "or" produces the union. The pattern holds every time: less-than becomes an intersection, greater-than becomes a union.

Work through three examples end to end. First, solve 3x + 2 > 8 and x − 1 ≤ 4. The first gives x > 2, or (2, ∞). The second gives x ≤ 5, or (−∞, 5]. The word "and" means intersect, so the answer is (2, 5]. Draw the line, shade (2, ∞) and (−∞, 5], and the overlap is (2, 5].

Second, solve 2x − 1 < 3 or 4 − x ≥ 2. The first gives x < 2, or (−∞, 2). The second gives x ≤ 2, or (−∞, 2]. The word "or" means union. The union of (−∞, 2) and (−∞, 2] is just (−∞, 2], because the second interval already contains every point of the first. The answer simplifies. This shows that a union is not always longer; sometimes one interval swallows the other.

Third, find the domain of f(x) = 1/√(x² − 9). The radicand must be strictly positive, so x² − 9 > 0, which factors to (x − 3)(x + 3) > 0. The critical points are −3 and 3. Test x = −4, x = 0, and x = 4. The expression is positive for x < −3 and x > 3, negative between −3 and 3. The domain is (−∞, −3) ∪ (3, ∞). The union is required because the two regions are separate. The intersection of the two conditions x > 3 and x < −3 would be ∅, which is not the answer here.

These three examples cover the three typical results: a single bounded interval from an intersection, a single unbounded interval from a swallowing union, and a two-part unbounded interval from a disjoint union. The number line diagrams for each are the same in structure: shade the relevant regions, then read the answer from the shading. If you can draw the line, you can write the interval.

How to write interval notation is a matter of a few fixed rules. Always write the smaller endpoint on the left. Use a bracket [ or ] when the endpoint is included, a parenthesis ( or ) when it is excluded. Use ∞ or −∞ with a parenthesis only. Separate the two endpoints with a comma. If the set has multiple intervals, join them with ∪. If the set is empty, write ∅, never [ ]. The empty set is not an interval. It has no endpoints, so the notation for an interval cannot represent it.

A common trap is writing (a, b) and forgetting that this means all real numbers between a and b, not the ordered pair. In interval notation, (2, 5) is a set of numbers, not a point on a plane. This ambiguity is resolved by context, but in a guide about intervals, the set meaning is the only one in play. Similarly, {2, 5} means a set with exactly two elements, 2 and 5, which is not the same as [2, 5]. The bracket form includes every number between 2 and 5, which is infinitely many.

Another rule: never use a union symbol with a single interval. If the answer is just [1, 5], do not write [1, 5] ∪ ∅. The empty set adds nothing, and the expression looks like you are unsure. Likewise, do not write [1, 5] ∪ [2, 3] when you mean [1, 5]. The simplified form is the correct one. Simplification is part of how to write interval notation; the goal is the shortest correct expression.

Interval notation ranges meanings are fixed by convention. A closed interval [a, b] includes a and b. An open interval (a, b) excludes both. A half-open interval [a, b) includes a but not b. An unbounded interval (a, ∞) has no upper limit. These meanings are universal; there is no regional variation. OpenStax College Algebra 2e, section 1.1, defines all of these. The meaning of ∪ and ∩ is equally fixed. The union is the set of elements in either interval, and the intersection is the set of elements in both.

Knowing these meanings lets you read any interval notation without solving anything. If you see (−∞, 0) ∪ [0, ∞), you know immediately that the set is all real numbers except 0, which is why it is the domain of 1/x. If you see (−∞, ∞), that is the entire real line, often written as ℝ. The two are interchangeable, though (−∞, ∞) is more explicit about the interval structure. The empty set ∅ is the opposite of the full line, containing no numbers at all.

One more meaning to internalize: a single point is an interval. The set {3} is written as [3, 3], because that would include only the number 3, which is technically an interval of length zero, and most courses do use that form. Instead, use set-builder notation or just the braces. The complement of {3} is (−∞, 3) ∪ (3, ∞), which is a union of two intervals. This is the standard way to express "all real numbers except 3" in interval notation.

Compound inequalities interval notation is a related but distinct topic. It covers the same conversion from "and" and "or" statements into intervals, but it focuses on the inequality solving itself rather than the notation. The method is the same: solve each part, then combine. The notation does not change. If you can handle a single inequality, you can handle a compound one, provided you remember the union and intersection rules.

The most common mistakes in interval notation are all avoidable. Writing (5, 2) instead of (2, 5) is a wrong-order error; the left endpoint must be smaller. Omitting the ∪ between disjoint intervals, as in (−∞, 2)(3, ∞), is a missing-union error. Using brackets when the inequality is strict, writing [a, b] for a < x < b, is a double-inclusion error. Writing [ ] for the empty set is a notation error. And using ∪ when the intervals overlap is an overuse error. Each of these is a slip, not a conceptual failure, and each is caught by a quick check on a number line.

Another frequent issue is confusing the empty set ∅ with the number 0. They are unrelated. ∅ means no real numbers satisfy the condition; 0 is a specific number that may or may not be in a set. For example, the intersection [1, 2] ∩ [3, 4] is ∅, not 0. The number 0 is not between 1 and 2, nor between 3 and 4, so it is not in either interval, but that is irrelevant. The set is empty because no number is in both intervals.

To avoid these errors, always draw the number line. Shade the intervals, then look. If you are taking a union, the answer is every shaded region. If you are taking an intersection, the answer is only where the shading overlaps. If the overlap is empty, write ∅. This visual method is faster than algebra and nearly impossible to get wrong. It also shows you when a union simplifies to a single interval, which is the overuse error.

One more check: test a number. Pick a number from each interval you wrote and plug it back into the original inequality. If it fails, your notation is wrong. This is the final safety net. It takes a minute but catches every type of error, from a flipped bracket to a missing union. Do this on an exam, and you will never lose a point to a notation slip.

Domain and range are written in interval notation, and getting the union and intersection right is what separates a correct answer from an incorrect one. The domain of a square root function requires the radicand to be ≥ 0, which often produces a single closed interval. The domain of a logarithmic function requires the argument to be > 0, which produces an open interval. The domain of a rational function excludes zeros of the denominator, which produces a union of intervals. Every one of these is an exercise in union and intersection.

In calculus, intervals of increase or decrease are usually open intervals, because the derivative is zero at a local maximum or minimum. Describing these requires writing the union of several open intervals. For example, a cubic function might increase on (−∞, −1) and (1, ∞), which is written as a union. Getting the union wrong means describing the wrong behavior. This is not a pedantic detail; it is the difference between a correct graph sketch and a misleading one.

The same logic applies to range. The range of a function is an interval or a union of intervals. If a function has a hole, the range may be missing a single value, which you express as a union. For instance, the range of f(x) = (x² − 1)/(x − 1) is all real numbers except 2, written as (−∞, 2) ∪ (2, ∞). This is the same notation as a domain with a single excluded point. The skill transfers directly.

Many graphing calculators and computer algebra systems output interval notation for solutions. They show unions as a list of intervals separated by ∪, and they show intersections as a single interval or the symbol ∅. If you enter an inequality and the output is (−∞, 2) ∪ (3, ∞), that is the answer. If the output is ∅, no real numbers satisfy the condition. Trust the calculator's notation, but verify the endpoints. A calculator cannot tell you whether a bracket or parenthesis is correct; that depends on the inequality's strictness.

One caveat: some calculators require you to enter intervals in a specific format, and the output may not match the notation in your textbook. The mathematics is the same, but the syntax differs. Read the manual or the help text. If the calculator returns something like (−∞,2)U(3,∞), the U stands for union. If it returns an empty set symbol, that is ∅. Do not confuse a zero with the empty set; they are different keys on most devices.

The empty set is not a failure of notation; it is a legitimate answer. When an inequality has no solution, or when two intervals do not intersect, the answer is ∅. For example, the intersection [1, 2] ∩ [3, 4] is ∅, because no number is in both. The union [1, 2] ∪ [3, 4] is not empty; it is two separate intervals. Knowing when to write ∅ and when to write a union is a matter of reading the operation.

Some students write { } for the empty set, which is acceptable, but ∅ is more standard. Do not write [ ] or ( ) with nothing inside; those are not standard notation and will confuse a grader. The empty set is not the same as the interval (0, 0), which is itself empty because no number is both greater than 0 and less than 0. The notation (0, 0) is technically an empty interval, but it is unconventional. Use ∅ for clarity.

In a real-world context, the empty set appears when a function has no valid input. For example, the domain of f(x) = √(x) with x restricted to negative numbers is ∅, because the square root of a negative number is not real. The domain of f(x) = 1/√(x² + 1) is all real numbers, because the radicand is always positive. The difference is the presence or absence of a solution, and interval notation captures it cleanly.

You now have every tool to combine intervals correctly. Identify the operation: ∪ for "or", ∩ for "and". Draw the number line. Shade the intervals. Read the result. If the result has multiple pieces, join them with ∪. If it has none, write ∅. Check your endpoints against the original inequalities. Test a number from each interval. This sequence takes two minutes and produces a correct answer every time.

The only remaining risk is overconfidence. Interval notation is unforgiving of a single misplaced bracket. A parenthesis instead of a bracket at the wrong endpoint changes the answer from including a number to excluding it. The difference between [2, 5] and (2, 5) is the number 2. That is not a trivial distinction; it is the difference between a function being defined at 2 or not. Precision is the entire point of the notation.

Practice on paper, not in your head. The number line is a crutch that you should not discard until the notation feels automatic. Even experienced mathematicians draw a line for a complicated union. The visual is not a sign of weakness; it is a guarantee against error. Use it every time, and you will never write a wrong interval again.