Less Than or Equal To — Definition, Formula & Examples
Less than or equal to (≤) is a comparison symbol that means one value is either smaller than or exactly equal to another value. For example, writing means can be 5, or any number below 5.
The relation holds if and only if or . Equivalently, means there is no case in which is greater than . The symbol ≤ combines the strict inequality < with equality = into a single weak inequality.
Key Formula
Where:
- = The value on the left side of the comparison
- = The value on the right side of the comparison
How It Works
You use ≤ whenever a value is allowed to match the boundary, not just be below it. If a roller coaster sign says "riders must be ≤ 200 pounds," someone who weighs exactly 200 pounds is still allowed. On a number line, includes 4 and every number to its left — you show this with a filled-in dot at 4. This differs from strict less than (), where 4 itself is not included and you use an open dot. When solving inequalities, you can add, subtract, multiply, or divide both sides just like an equation, but remember to flip the ≤ symbol if you multiply or divide by a negative number.
Worked Example
Problem: Solve the inequality 2x + 3 ≤ 11 and list three whole-number solutions.
Step 1: Subtract 3 from both sides to isolate the term with x.
Step 2: Divide both sides by 2. Since 2 is positive, the direction of ≤ stays the same.
Step 3: Identify solutions. Any number that is 4 or less works. Three whole-number solutions are 4, 3, and 0.
Answer: The solution is . Notice that is included because the symbol is ≤, not <.
Another Example
Problem: A student can bring at most $10 to the school book fair. Write an inequality for the amount of money m the student can bring, then decide whether $10 is allowed.
Step 1: "At most $10" means the amount can equal 10 or be less than 10. Write this as an inequality.
Step 2: Check whether m = 10 satisfies the inequality. Since 10 ≤ 10 is true (10 equals 10), the student can bring exactly $10.
Answer: The inequality is , and yes, bringing exactly $10 is allowed.
Why It Matters
You will use ≤ constantly in pre-algebra and algebra courses whenever problems involve phrases like "at most," "no more than," or "up to." Real-world constraints — speed limits, budget caps, weight limits on elevators — are naturally expressed with ≤. Mastering this symbol prepares you for graphing inequalities on number lines and on the coordinate plane.
Common Mistakes
Mistake: Confusing ≤ with <, and excluding the boundary value.
Correction: The line under the symbol means "or equal to." With , the value 4 is a valid solution. Use a filled-in dot on the number line to remind yourself the endpoint is included.
Mistake: Forgetting to flip ≤ to ≥ when multiplying or dividing both sides by a negative number.
Correction: Multiplying or dividing an inequality by a negative number reverses the order. For example, from , dividing by gives , not .