What Is an Inequality
An inequality is basically a comparison between two things—numbers, variables, or expressions—that aren’t necessarily equal. Sometimes they can be equal… and that depends on the symbol you’re using. You’ll see that in a second.
Equation vs. Inequality
Let’s start with something that’s not an inequality: x = 10. That’s an equation. And here’s the key: an equation like this has one exact solution. The only way it’s true is if x is 10. If you try x = 9, you’d be saying 9 = 10, which is false. So for x = 10, the solution set is basically just 10.
Now inequalities behave differently. The five inequality symbols you’ll see all the time. Here they are:
| Symbol | Meaning |
| ≠ (Not equal to) | Values are not the same |
| > (Greater than) | Value is strictly greater |
| < (Less than) | Value is strictly less |
| ≥ (Greater than or equal to) | Value is greater or equal |
| ≤ (Less than or equal to) | Value is less or equal |
Note, one really helpful tip:
The “open/wider” side of the symbol points toward the larger value. So if it’s opening toward 10, that side is the bigger side.
Inequality Examples
What each one actually means (using 10 as the anchor)
1) x > 10 (x is greater than 10)
This means any number bigger than 10 works. So 11 works, 50 works, and 150 works. You could keep going forever—there are infinitely many solutions. But 10 does NOT work, because 10 is not greater than 10.
2) x ≥ 10 (x is greater than or equal to 10)
This one is almost the same as x > 10, but now 10 is allowed. So everything greater than 10 still works (11, 50, 150…), and 10 works too, because it’s “equal to” 10. That little “or equal to” part is a big deal. It means the boundary value is included.
3) x < 10 (x is less than 10)
Now we’re going the other direction. Any number smaller than 10 works: 1, 2, 8, -8… whatever, as long as it’s less than 10. Again: infinitely many solutions. And again: 10 does NOT work, because 10 is not less than 10.
4) x ≤ 10 (x is less than or equal to 10)
Same idea as < 10, except now the boundary value is included. So all numbers less than 10 work… and 10 works too.
5) x ≠ 10 (x is not equal to 10)
This one’s super straightforward. Anything that’s not 10 is a solution:
8, 9, -20, 1000… all good. The only thing that doesn’t work is 10, because 10 is definitely equal to 10.
Try a few like the examples
Example A: 7 < a
This reads: “7 is less than a.” So a has to be bigger than 7. Valid choices: 8, 52, 100… endless options. But 7 is not included because there’s no “or equal to.”
Example B: 11 ≥ b
This reads: “11 is greater than or equal to b.” That means b can be 11 or anything smaller. So 11 works, 3 works, -1 works… anything ≤ 11.
Example C: c < 5
That means c must be less than 5. So 1, 0, -9 all work. But 5 does not, because it’s strictly less than.
Compound Inequalities
Now for an "upgraded" version: 3 < n < 7
This can be read as: "3 is less than n, and n is less than 7." A more intuitive meaning is: n is between 3 and 7. n represents a range of values.
Final Thought
So you see, inequalities aren't scary at all. They are simply used to: compare magnitudes (who is larger or smaller) or describe conditions/ranges (what is the allowed interval).
