No 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, ≥.
No more than 20 includes 20 itself, so the dot is closed and the symbol is ≤. Read the whole phrase: the word more does not mean > here.
Formal definition and notation
The phrase a quantity x is no more than c states that x does not exceed c: x ≤ c, the negation of x > c. Its solution set includes the boundary value c, shown on a number line by a closed dot at c and shading to the left, and in interval notation as (−∞, c], possibly further limited by the context (for example, x ≥ 0 for counts).
Symbol
Read it as
Meaning
≤
is less than or equal to
no more than, at most, not more than, a maximum of
≥
is greater than or equal to
no less than, at least, a minimum of
Key facts
No more than c is ≤ c: the value cis included.
Think of it as a ceiling: you can go up to the limit and touch it, but not past it.
More than c is > c, and no more than c is the exact opposite: every number is one or the other.
Word order can flip a sentence: the cost is no more than 50 is C ≤ 50, and 50 is no more than the cost is 50 ≤ C.
For whole items (people, boxes), the answer to a no-more-than problem is the largest whole number that fits, so round down.
Try it: no more than 6
Interactive
The shading shows x ≤ 6: no more than 6. Drag the endpoint to move the limit, and use the button to make the right end open to see how no more than differs from less than.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
No More Than examples, step by step
Example 1: Translate the limit
Warm-up
An airline says a carry-on bag can weigh no more than 22 pounds. Write an inequality for the weight w.
Because a weight cannot be negative, the full answer is 0 ≤ w ≤ 22.
Answer:w ≤ 22 pounds (with w ≥ 0).
Example 2: Elevator limit
Core
An elevator can carry no more than 1,200 pounds. A worker weighing 180 pounds rides with boxes that weigh 140 pounds each. What is the greatest number of boxes he can take?
Inequality: 180 + 140b ≤ 1200.
Subtract 180: 140b ≤ 1020.
Divide by 140: b ≤ 7.28….
Boxes are whole, so round down: 8 boxes would weigh 180 + 1120 = 1300, too much.
Answer: He can take at most 7 boxes.
Example 3: Largest possible number
Test-style
The sum of a number n and 9 is no more than 25. What is the greatest possible value of n?
Write it: n + 9 ≤ 25.
Subtract 9: n ≤ 16.
16 is allowed because of the equal part of ≤: 16 + 9 = 25.
Answer: The greatest possible value is n = 16.
Common mistakes
✗ MistakeYou can spend no more than 30 dollars: x < 30.
✓ Insteadx ≤ 30. Spending exactly 30 dollars is allowed.
No more than means the limit itself is fine. Use < only for phrases like less than or fewer than.
✗ MistakeNo more than 12 students: s > 12, because of the word more.
✓ Insteads ≤ 12.
The word no reverses more. Translate the whole phrase, not one keyword.
✗ MistakeA van holds no more than 1,000 lb; boxes weigh 300 lb, so b ≤ 3.33 means 4 boxes.
✓ InsteadRound down: 3 boxes. Four boxes weigh 1,200 lb.
With a maximum, rounding up breaks the limit. Check your whole-number answer in the original inequality.
Less than (<) leaves the boundary out: fewer than 5 cookies means 0 to 4. No more than 5 includes 5.
No More Thanvs.No less than
No less than c is a floor, x ≥ c: you can be at c or above. No more than c is a ceiling, x ≤ c.
Quick check: 3 questions
Question 1 of 3
A recipe uses no more than 3 cups of sugar. Which inequality fits?
Show answer and explanation
Answer: D. No more than includes 3 and anything below it: s ≤ 3. A leaves out 3, and B and C point the wrong way.
Question 2 of 3
Which number is NOT a solution of: x is no more than −2?
Show answer and explanation
Answer: C.x ≤ −2. Zero is greater than −2, so it fails. −2 itself works, and −5 and −2.5 are farther left on the number line.
Question 3 of 3
A phone plan costs 15 dollars plus 0.10 dollars per text, and the bill must be no more than 40 dollars. What is the greatest number of texts?
Show answer and explanation
Answer: B.15 + 0.10t ≤ 40, so 0.10t ≤ 25 and t ≤ 250. 251 breaks the limit, 400 ignores the 15 dollars, and 550 adds the 15 instead of subtracting.
No More Than: frequently asked questions
Does no more than include the number?
Yes. No more than 10 means 10 is allowed, along with every value below it, so you write x ≤ 10 and draw a closed dot at 10. Only phrases such as less than or fewer than leave the number out. A good test: could the amount be exactly 10 without breaking the rule? If yes, use ≤.
Is no more than the same as at most?
Yes. No more than, at most, not more than, a maximum of and up to (when the end is included) all translate to ≤. Tests use these phrases interchangeably, so learn them as a group, along with their opposites: no less than, at least and a minimum of, which translate to ≥.
What is the symbol for no more than?
The symbol is ≤, read is less than or equal to. The bar under the angle shows that equality is allowed. On a number line, it is a closed (filled) dot at the limit with shading toward smaller numbers, and in interval notation the limit gets a square bracket.
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