Greater than or equal to means a value is either bigger than a number or exactly equal to it, and its symbol is ≥. The inequality x ≥ 3 is true for 3 and every number above it. It means “at least 3”. On a number line it is graphed with a filled dot, because the endpoint is included.
In Spanish
mayor o igual que
Also called
greater than or equal to sign, at least, no less than, minimum, ≥
The only difference between x ≥ 3 and x > 3 is the number 3 itself: the filled dot puts it in the solution, the open circle leaves it out.
Formal definition and notation
For real numbersa and b, a ≥ b means a > b or a = b; equivalently, a − b ≥ 0, so a − b is positive or zero. It is the same statement as b ≤ a. The solution set of x ≥ c is the closed ray of all numbers from c to the right, written [c, ∞) in interval notation.
how calculators, spreadsheets and code type the symbol
Key facts
The bar under the symbol is half of an equals sign: ≥ is > and = combined.
Boundary included:5 ≥ 5 is true, while 5 > 5 is false.
Graph it with a filled (closed) dot at the boundary and shade or draw an arrow to the right.
Word clues: at least, no less than, a minimum of, not below. “You must be at least 16” means age ≥ 16.
Solve it like an equation, but if you multiply or divide both sides by a negative number, flip it to ≤: −2x ≥ 6 gives x ≤ −3.
Try it: graph x ≥ a
Interactive
The filled dot is included, and the arrow shows that every larger number works too. Drag the dot, or press Left end: open/closed to compare ≥ with >.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Greater Than or Equal To examples, step by step
Example 1: Test values
Warm-up
Which of −1, 4, 4.5, 7 make x ≥ 4.5 true?
Check each number against 4.5. −1 and 4 are less than 4.5, so they fail.
4.5 ≥ 4.5 is true, because the numbers are equal.
7 ≥ 4.5 is true, because 7 is greater.
Answer: 4.5 and 7 make it true.
Example 2: Solve and graph
Core
Solve 3x − 5 ≥ 13 and describe the graph.
Add 5 to both sides: 3x ≥ 18.
Divide both sides by 3 (a positive number, so the symbol stays): x ≥ 6.
Check the boundary: 3(6) − 5 = 13, and 13 ≥ 13 is true, so 6 belongs.
Graph: a filled dot at 6 and an arrow to the right.
Answer:x ≥ 6: a filled dot at 6 with the line shaded to the right.
Example 3: At least, in a word problem
Test-style
A club needs at least 150 dollars for a trip. It has 62 dollars and earns 8 dollars for each car it washes. What is the least number of cars it must wash?
“At least” means ≥: 62 + 8c ≥ 150.
Subtract 62: 8c ≥ 88.
Divide by 8: c ≥ 11.
Check: 11 cars give 62 + 88 = 150 dollars, which meets the goal exactly.
Answer: The club must wash at least 11 cars.
Common mistakes
✗ MistakeGraphing x ≥ −2 with an open circle at −2.
✓ InsteadUse a filled dot at −2, because −2 ≥ −2 is true.
An open circle means the boundary is left out, which is the graph of > instead of ≥.
✗ Mistake−4x ≥ 12, so x ≥ −3.
✓ Instead−4x ≥ 12, so x ≤ −3.
Dividing both sides by a negative number reverses the order. Test x = −5: −4(−5) = 20 ≥ 12, true, and −5 ≤ −3.
✗ Mistake“At least 18” written as x ≤ 18.
✓ Instead“At least 18” is x ≥ 18; “at most 18” is x ≤ 18.
“At least” sets a minimum, so the allowed values are 18 and above.
≤ allows the number or anything below it. x ≤ 3 means “at most 3”; x ≥ 3 means “at least 3”.
Quick check: 3 questions
Question 1 of 3
Which number is not a solution of n ≥ −2?
Show answer and explanation
Answer: C.−3 is left of −2, so −3 ≥ −2 is false. −2 works because it is equal, and 0 and 5 are greater.
Question 2 of 3
A ride requires riders to be at least 48 inches tall. Which inequality fits?
Show answer and explanation
Answer: A. “At least 48” allows exactly 48 and anything taller, so h ≥ 48. h > 48 would turn away a rider who is exactly 48 inches; the other two describe a maximum.
Question 3 of 3
Solve −2x + 1 ≥ 9.
Show answer and explanation
Answer: D. Subtract 1: −2x ≥ 8. Divide by −2 and flip the symbol: x ≤ −4. x ≥ −4 forgets to flip, and x ≤ −5 adds 1 instead of subtracting.
Greater Than or Equal To: frequently asked questions
How do you type the greater than or equal to symbol?
In most calculators, spreadsheets and programming languages, type a greater-than sign followed by an equals sign: >=. In word processors you can insert the single symbol ≥ from the symbol menu; on Windows, Alt+242 works on many keyboards. In handwritten math, draw > with a short line underneath.
Does greater than or equal to use an open or closed circle?
A closed, filled circle. The filled dot shows that the boundary number is part of the solution, since x ≥ 3 is true when x = 3. Strict inequalities, > and <, use an open circle because the boundary is not included. The arrow or shading then runs to the right for ≥.
Is "at least" greater than or equal to?
Yes. “At least 10” means 10 or more, which is x ≥ 10. Other phrases with the same meaning are “no less than”, “a minimum of” and “not below”. Do not mix it up with “at most”, which means ≤, or “more than”, which means strictly >.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.EE.B.8, CCSS 7.EE.B.4.b
Cite this entry
Effortless Math Team. “Greater Than or Equal To.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/greater-than-or-equal-to/
Flashcard
TermGreater Than or Equal To≥
DefinitionBigger than or exactly equal to; “at least”. x ≥ 3 includes 3, so graph it with a filled dot at 3.
Real 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 1NumberA 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 2Interval 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, PrecalculusInequalityAn 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 8SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3SideA 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 2Negative 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 7CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7Greater ThanGreater than means that one number is bigger than another, and its symbol is >. The statement 7 > 3 reads “seven is greater than three”. The open side of the symbol faces the bigger number and the point aims at the smaller one. On a number line, the greater number is always farther to the right.Symbols and Notation · Grade 1, Grade 2, Grade 3Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5Less 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 1Number 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 6Compound InequalityA 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.Algebra · Algebra 1, Algebra 2SolutionA 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 8Less ThanLess than means smaller in value, and its symbol is <. The statement 4 < 9 reads “4 is less than 9.” The pointed end of the symbol always faces the smaller number. On a number line, the smaller number is farther left, so −3 < 2 even though 3 is bigger than 2. Less than does not include equal.Symbols and Notation · Grade 1, Grade 2, Grade 6At 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 1EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8