Less 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.
In Spanish
menor o igual que
Also called
≤, at most, no more than, less than or equal to sign, <=
The only difference between ≤ and < is the boundary number. With less than or equal to, the boundary counts, so its dot is filled in.
Formal definition and notation
For real numbersa and b, a ≤ b means a < b or a = b; equivalently, b − a ≥ 0. The solution set of x ≤ c is every real number up to and including c. On a number line it is drawn with a closed (filled) dot at c and shading to the left; in interval notation it is (−∞, c], where the square bracket shows that c is included. An inequality with ≤ keeps its direction when you add, subtract, or multiply or divide by a positive number, and reverses to ≥ when you multiply or divide by a negative number.
Symbol
Read it as
Meaning
≤
is less than or equal to
the left side is at most the right side; also typed as <= in spreadsheets and code
interval notation for x ≤ 5; the bracket means 5 is included
Key facts
The equal part matters: 5 ≤ 5 is true, while 5 < 5 is false.
Word phrases for ≤: at most, no more than, not more than, a maximum of, up to, does not exceed.
Graph x ≤ c with a closed dot at c. Graph x < c with an open dot.
Reading right to left flips the words: 3 ≤ x says the same thing as x ≥ 3.
Multiplying or dividing both sides by a negative number reverses the symbol: −x ≤ 2 becomes x ≥ −2.
An integer answer may need rounding down: if n ≤ 29.3 and n counts whole items, the largest choice is 29.
The flip rule for ≤
The flip rule for ≤
If a ≤ b and c < 0, then ac ≥ bc
a ≤ b
the inequality you start with
c
a negative number you multiply or divide both sides by
Example: −2x ≤ 8. Divide both sides by −2 and flip the sign: x ≥ −4. Check with x = 0: −2(0) = 0 ≤ 8, true.
Try it: open or closed?
Interactive
Drag the two endpoints, then press the open/closed buttons. A filled end is part of the inequality (≤); an open end is not (<). The readout writes the inequality and its interval notation.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Less Than or Equal To examples, step by step
Example 1: Test values
Warm-up
Which of the numbers −4, 0, 6 and 7.5 make x ≤ 6 true?
Divide both sides by −3 and flip the sign: x ≥ −4.
Check x = 0: −3(0) + 5 = 5 ≤ 17, true. Check x = −5: 20 ≤ 17, false.
Graph: a closed dot at −4 with shading to the right.
Answer:x ≥ −4: a closed dot at −4, shaded to the right.
Example 3: At most, in a word problem
Test-style
An elevator can carry at most 1,500 pounds. A 180-pound worker rides with boxes that weigh 45 pounds each. What is the greatest number of boxes the worker can take?
At most means ≤: 180 + 45b ≤ 1500.
Subtract 180: 45b ≤ 1320.
Divide by 45: b ≤ 29.3 (rounded).
Boxes come in whole numbers, and 30 boxes would break the limit, so round down.
Answer: The worker can take at most 29 boxes.
Common mistakes
✗ MistakeGraphing x ≤ 2 with an open circle at 2.
✓ InsteadUse a closed (filled) dot at 2 and shade to the left.
The equal part of ≤ means 2 is a solution, and a filled dot shows that the endpoint is included.
✗ MistakeWriting at most 10 as x ≥ 10.
✓ InsteadAt most 10 means x ≤ 10: 10 is the largest value allowed.
At most sets a ceiling, so the values must be below or at it. At least sets a floor and uses greater than or equal to.
✗ MistakeSolving −4x ≤ 20 as x ≤ −5.
✓ InsteadDividing by −4 flips the sign: x ≥ −5.
Check x = 0: −4(0) = 0 ≤ 20 is true, yet 0 is not ≤ −5. The flipped answer passes the check.
✗ MistakeCalling 5 ≤ 5 false because 5 is not less than 5.
✓ Instead5 ≤ 5 is true, because 5 equals 5.
The statement is true when either part holds: less than, or equal.
Less than leaves out the boundary: x < 5 is false for 5. Less than or equal to includes it: x ≤ 5 is true for 5. The graphs differ only in the dot at 5, open versus closed.
An inequality is any statement that compares with <, >, ≤, ≥ or ≠. Less than or equal to is one of those symbols.
Quick check: 3 questions
Question 1 of 3
Which number is not a solution of x ≤ −2?
Show answer and explanation
Answer: D. 0 is greater than −2, so 0 ≤ −2 is false. −2 works because it is equal, and −5 and −2.5 are both less than −2.
Question 2 of 3
A score s of at most 80 points. Which inequality matches?
Show answer and explanation
Answer: A. At most 80 means 80 is allowed and nothing higher is, so s ≤ 80. s < 80 wrongly leaves out 80, and the other two describe at least or more than 80.
Question 3 of 3
Solve 2x − 7 ≤ 9.
Show answer and explanation
Answer: C. Add 7: 2x ≤ 16. Divide by 2, a positive number, so the sign stays: x ≤ 8. x ≤ 1 subtracts 7 instead of adding, x ≥ 8 flips the sign without a reason, and x < 8 drops the equal part.
Less Than or Equal To: frequently asked questions
Is 5 less than or equal to 5?
Yes. Less than or equal to is true when the first number is smaller or when the two are equal, so 5 ≤ 5 is a true statement. That is the key difference from plain less than: 5 < 5 is false. On a graph, this is why x ≤ 5 uses a filled dot at 5.
What does at most mean in math?
At most means no more than, so it uses ≤. A bag that holds at most 12 apples can hold 12 or any smaller number, but not 13: a ≤ 12. Phrases like a maximum of, up to and does not exceed mean the same thing. Its opposite, at least, uses ≥.
How do you type the less than or equal to symbol?
On a Mac, press Option and the comma key. In Microsoft Word, type 2264 and then press Alt and X together. On a phone, press and hold the < key on the symbol keyboard. In spreadsheets, calculators and programming languages, type the two characters <= instead.
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. “Less Than or Equal To.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/less-than-or-equal-to/
Flashcard
TermLess Than or Equal To≤
DefinitionSmaller than or exactly equal to; means at most. x ≤ 3 includes 3, so graph it with a closed dot at 3.
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Numbers include whole numbers, fractions, decimals and negatives.Number System · 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 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 2Compound 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 2No More ThanNo more than means less than or equal to, written ≤. The limit itself is allowed: you may spend no more than 20 dollars means 20 dollars or any smaller amount, so x ≤ 20. It means the same as at most. The reverse phrase, no less than, means greater than or equal to, ≥.Problem Solving · Grade 6, Grade 7, Algebra 1RoundingRounding replaces a number with a nearby number that is easier to use, to a chosen place value. Look at the digit immediately to the right of that place: 5 or more rounds up, 4 or less rounds down. So 47 rounds to 50 to the nearest ten, and 3.68 rounds to 3.7 to the nearest tenth.Number System · Grade 3, Grade 4, Grade 5Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. 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To solve one, identify what the problem is asking you to find, translate the words into an expression or equation, solve it, and check that the answer makes sense in the story. It is also called a story problem or a real-world problem.Problem Solving · Grade 3, Grade 4, Grade 5Greater 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 3At 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 1