A compound inequality is two inequalities joined by “and” or “or”. With “and”, a number must make both true, as in −2 < x ≤ 3, which is one segment on a number line. With “or”, it only has to make at least one true, as in x < −1 or x ≥ 4, which is often two separate pieces.
In Spanish
desigualdad compuesta
Also called
double inequality, and inequality, or inequality, combined inequality
An and inequality keeps only the numbers that work for both parts, so it is one piece. An or inequality keeps every number that works for either part, so here it is two rays.
Formal definition and notation
A compound inequality joins two inequalities with a logical connective. The solution set of “A and B” is the intersection A ∩ B of the two solution sets; the solution set of “A or B” is their union A ∪ B. The double inequality a < x < b means a < x and x < b. An “and” inequality can have no solution, such as x < 1 and x > 4.
Or means either: x ≤ −3 or x > 2 includes −10 and 5 but not 0.
An open dot marks < or > (endpoint left out); a filled dot marks ≤ or ≥ (endpoint included).
Some “and” inequalities have no solution (x < 0 and x > 5), and some “or” inequalities include every number (x < 5 or x > 0).
Absolute value inequalities turn into compound ones: |x| < 3 means −3 < x < 3, and |x| > 3 means x < −3 or x > 3.
How to solve a compound inequality
How to solve a compound inequality
a < px + q < b ⇒ a − q < px < b − q
p, q
the numbers attached to x
a, b
the outer bounds
In a double inequality, do the same operation to all three parts. If you multiply or divide by a negative number, reverse both inequality signs. For an “or” inequality, solve each part separately.
Try it: build an and-inequality
Interactive
Drag the two endpoints and switch each end between open and closed. The readout writes the compound inequality and its interval notation. Send one end to infinity to see a single inequality such as x > −2.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Compound Inequality examples, step by step
Example 1: Solve a double inequality
Warm-up
Solve −3 < x + 2 ≤ 6.
Subtract 2 from all three parts: −3 − 2 < x + 2 − 2 ≤ 6 − 2.
Interval notation is a way to write a solution set, such as (−2, 3]. A compound inequality is the statement itself, −2 < x ≤ 3.
Compound Inequalityvs.System of inequalities
A compound inequality usually has one variable and its graph is on a number line. A system of inequalities usually has two variables and is graphed as a shaded region of the coordinate plane.
Quick check: 3 questions
Question 1 of 3
Which graph matches x ≤ −2 or x > 1?
Show answer and explanation
Answer: B. Or means either piece works: shade left from −2 (filled, because of ≤) and right from 1 (open, because of >). The segment is an and graph, and the third choice swaps the dots.
Question 2 of 3
Solve 1 ≤ 2x + 3 < 9.
Show answer and explanation
Answer: A. Subtract 3: −2 ≤ 2x < 6. Divide by 2: −1 ≤ x < 3. The second choice stops before dividing, and the others make an arithmetic slip on one side.
Question 3 of 3
Which compound inequality has no solution?
Show answer and explanation
Answer: D. No number is both less than 2 and greater than 5. The first choice includes every number, the second is 2 < x < 5, and the third is two rays.
Compound Inequality: frequently asked questions
What is the difference between and and or inequalities?
With and, a number must satisfy both inequalities, so the solution is the overlap of the two graphs, usually a single segment such as 1 < x < 6. With or, a number needs to satisfy only one of them, so the solution is everything either graph covers, often two rays pointing in opposite directions, such as x < 1 or x > 6.
How do you solve a compound inequality with three parts?
Treat it like an equation with three sides: whatever you do to the middle, do to the left and right too. For −3 < 2x + 1 ≤ 7, subtract 1 from all three parts and then divide all three by 2, giving −2 < x ≤ 3. If you divide by a negative number, reverse both inequality signs.
Can a compound inequality have no solution?
Yes, an and inequality can. x < −1 and x > 3 has no solution, because no number is in both places at once. An or inequality can go the other way and include every real number, such as x > −1 or x < 3, because every number satisfies at least one part.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-REI.B.3, CCSS HSA-CED.A.1
Cite this entry
Effortless Math Team. “Compound Inequality.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/compound-inequality/
Flashcard
TermCompound Inequality−2 < x ≤ 3
DefinitionTwo inequalities joined by and (overlap, one piece) or or (either part, often two pieces). −2 < x ≤ 3 means x > −2 and x ≤ 3.
InequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7, Grade 8Interval NotationInterval notation is a short way to write a set of numbers between two endpoints, using brackets and parentheses. A bracket, [ or ], means the endpoint is included; a parenthesis, ( or ), means it is not. So −2 ≤ x < 3 is written [−2, 3), and x > 4 is written (4, ∞).Symbols and Notation · Algebra 1, Algebra 2, PrecalculusNumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8System of EquationsA system of equations is a set of two or more equations that use the same variables and are solved together. Its solution is the values that make every equation true at the same time. For two linear equations, that is the point where the two lines cross. A system can have one solution, none, or infinitely many.Algebra · Grade 8, Algebra 1Domain and RangeDomain and range describe a function: the domain is the set of all inputs it accepts, and the range is the set of all outputs it produces. For the points (1, 4) and (2, 5), the domain is {1, 2} and the range is {4, 5}. Restrictions usually come from division by zero or square roots of negatives.Functions · Grade 8, Algebra 1, Algebra 2At Least and At MostAt least and at most are word-problem phrases for limits that include the boundary. “At least 3” means 3 or more, written x ≥ 3. “At most 3” means 3 or less, written x ≤ 3. In both cases the number 3 itself is allowed, so it gets a filled dot on a number line.Problem Solving · Grade 6, Grade 7, Algebra 1Less Than or Equal ToLess than or equal to is a comparison meaning one value is smaller than another or exactly equal to it, written with the symbol ≤. The statement x ≤ 5 is true for 5 itself and for every number below 5, such as 4.9, 0 and −3. In words, ≤ means at most or no more than.Symbols and Notation · Grade 6, Grade 7, Algebra 1