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Algebra › Inequalities
Inequalities
Sometimes the answer isn't one number — it's a whole range. Inequalities describe those ranges.
Shade an inequality
Pick the sign and the boundary. The shaded part is where x lives.
x ≤ 2
open — boundary excludedclosed — boundary included
Sign
Boundary2
The four signs
- < — less than
- > — greater than
- ≤ — less than or equal to
- ≥ — greater than or equal to
Reading them
x > 3 means: x is bigger than 3. So x could be 3.1, 4, 100, anything strictly more than 3 (but not 3 itself).
x ≤ −2 means: x is at most −2. So x can be −2, or smaller (−5, −10, −1000…).
Solving inequalities
Solve them like equations — same balance rules. With one twist:
The flip rule
When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. So
−2x < 6 becomes x > −3 (we divided by −2 and flipped).On a number line
- An open circle at the boundary means strictly less/greater (not including).
- A closed circle means ≤ or ≥ (the boundary is included).
- Shade the side where the variable lives.
Worked example
Solve 3x − 1 ≤ 11:
- Add 1:
3x ≤ 12. - Divide by 3 (positive — sign doesn't flip):
x ≤ 4. - On a number line: closed circle at 4, shade everything to the left.
Quick check
Solve −2x < 6.
Quick check
Which inequality matches a closed circle at 4 with shading to the left?