the whole idea, in one sentence
An inequality is just a balance scale where one side is heavier — and the same rules apply, with ONE twist: if you multiply or divide by a negative, flip the sign.
No wrong choice. Try one. Switch if it's not clicking.
You've got $20 in your pocket. You want to buy a sweatshirt, but you don't know the price yet — call it x.
You want the sweatshirt to cost less than or equal to what you have.
x ≤ 20
That's an inequality. It doesn't say "x equals 20" — it says "x is some number that's 20 or smaller."
Could be $20 (just barely makes it). Could be $19.99. Could be $5. Anything from there down works.
That's the difference: an equation has one answer ("x is exactly 20"). An inequality has a whole range of answers ("x is anywhere from negative-infinity up to 20").
The symbols:
< — less than (smaller, not equal)
> — greater than (bigger, not equal)
≤ — less than OR equal to (smaller, or exactly the same)
≥ — greater than OR equal to (bigger, or exactly the same)
You solve them the exact same way as equations — undo what's stuck to x, do it to both sides. With ONE exception, which we'll get to in a sec.
Inequalities live on a number line. The answer isn't one dot — it's a whole stretch of the line, shaded.
Take x < 3. That means "x is any number smaller than 3."
←———————○———→
··· -1 ·· 0 ·· 1 ·· 2 ·· 3
open circle at 3 (3 NOT included) · shade everything to the left
The circle is open (hollow) because 3 itself doesn't count — "less than 3" means smaller, not equal.
Now take x ≥ 1. "x is 1 or anything bigger."
←———●———————→
··· -1 ·· 0 ·· 1 ·· 2 ·· 3 ···
filled circle at 1 (1 IS included) · shade everything to the right
The circle is filled because 1 itself counts ("greater than OR equal to" — the "or equal to" part includes it).
Rule of thumb:
< or > → open circle (don't include the endpoint)
≤ or ≥ → filled circle (include the endpoint)
Solving an inequality is the same recipe as solving an equation. Undo what's stuck to x, do it to both sides.
Example: 2x + 1 < 9
2x + 1 < 9
Step 1. Subtract 1 from both sides.
2x < 8
Step 2. Divide both sides by 2.
x < 4
Done. The answer is "x is any number less than 4."
⚠️ THE ONE TWIST. If you multiply or divide both sides by a negative number, you have to flip the inequality sign.
Example: -3x > 9
-3x > 9
Divide both sides by -3. Because we divided by a negative, the > flips to <:
x < -3
Why? Quick proof: 2 is bigger than 1, right? But multiply both by -1, and you get -2 and -1. Now -2 is SMALLER than -1. Negatives flip the order. That's all. Don't forget the flip.
Check your answer: pick any number from your answer range and plug it back in. If it makes the original true, you got it.
5 problems. Type your answer using format like x<7 or x≥4 (no spaces). For the word problem, just enter the number.