How can I represent a range of possible values with an inequality?
Quick answer
Use an inequality to describe every value that meets a comparison. The symbols < and > exclude the boundary value; ≤ and ≥ include it. For example, x > 3 means all real numbers greater than 3, while x ≤ 1 means all real numbers up to and including 1. You can also show the range on a number line or in interval notation. [S1]
The question, the details, the next step.
I have a condition like “the number is greater than 3.” How do I write that as an inequality, and does it mean just one answer?
Write it as x > 3, where x stands for the number you are considering. The symbol > means “is greater than,” so 4, 5, and 3.1 all satisfy the condition. The inequality describes the whole set of qualifying values, not one particular answer. Over the real numbers, x > 3 is represented in interval notation as (3, ∞). The parenthesis at 3 signals that 3 itself is not included. [S1]
To check a proposed value, substitute it for x and ask whether the comparison is true: 4 > 3 is true, but 3 > 3 is false.
I mix up the four symbols. Can you give me a compact reference, especially for whether the endpoint counts?
Here is a symbol-and-meaning reference:
x < a: x is less than a; a is excluded.x ≤ a: x is less than or equal to a; a is included.x > a: x is greater than a; a is excluded.x ≥ a: x is greater than or equal to a; a is included.
For example, x ≤ 1 includes 1 and every real number below it. Its interval notation is (−∞, 1]; the square bracket at 1 shows that the endpoint is included. [S1]
A quick memory aid: the extra line in ≤ or ≥ adds “or equal to,” so the boundary value counts. Without that line, the comparison is strict and the boundary does not count.
How do I draw one on a number line so someone can see the range at a glance?
Mark the boundary number first. Use an open circle when the boundary is excluded (< or >), and a filled circle when it is included (≤ or ≥). Then shade in the direction of the values that make the inequality true: left for values less than the boundary, right for values greater than it. [S1]
Worked example: for x ≥ 2, put a filled circle at 2 and shade to the right. The filled circle includes 2; the shading represents values larger than 2. For x < 2, use an open circle at 2 and shade left. In either case, testing the boundary itself helps catch a mistaken open or filled circle.
Could you show how to choose the right form from a sentence and check my answer?
Try this sentence: “A container can hold at most 8 units.” Let c represent the amount it holds. “At most 8” includes 8 and allows smaller amounts, so write c ≤ 8. On a number line, mark 8 with a filled circle and shade left. In interval notation, the range is (−∞, 8] if c is allowed to be any real number no greater than 8. [S1]
To check, test the boundary and a value on each side: 8 satisfies c ≤ 8, 7 satisfies it, and 9 does not. This checks the comparison, though a real situation may impose additional restrictions—for example, the sentence alone does not tell us whether negative amounts make sense. Keep such context limits separate from what the inequality itself states.
A practical checklist
Choose a letter for the value you are describing.
Translate the comparison phrase into a symbol: “less than,” “at least,” or another precise phrase.
Decide whether the boundary value is included; use ≤ or ≥ when it is.
Test the boundary and one value on either side to check the direction and endpoint.
If useful, show the same range on a number line or in interval notation.
A few more useful answers
What is the difference between `x > 3` and `x ≥ 3`?
x > 3 excludes 3, while x ≥ 3 includes it. On a number line, that means an open circle for the first and a filled circle for the second.
Does an inequality always describe infinitely many values?
Not necessarily in every setting: the allowed values depend on the context or domain. Over the real numbers, a one-sided inequality such as x > 3 describes all real numbers above 3.
What does infinity mean in interval notation?
It indicates that the range continues without a finite endpoint in that direction. Infinity is not a value included in the range, so interval notation uses a parenthesis beside it.
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra 2e: Solve Linear Inequalities openstax.org · Checked