Algebra
Single-variable inequalities
Solving inequalities
20 practice questions
2 video lessons
Theory + worked examples
Theory
Solve an inequality exactly like an equation, with one crucial exception:
Flip the inequality sign whenever you multiply or divide both sides by a negative number.
Adding, subtracting, or working with positives does not flip the sign.
Same as an equation, but flip for a negative.
\(x<4\) on a number line.
The flip rule:
\[-3x>12\ \Rightarrow\ x<-4\]
Only a negative multiplier/divisor flips the sign.
How to solve an inequality
- Use inverse operations to isolate the variable.
- Do the same to both sides.
- Flip the sign if you multiply/divide by a negative.
- Graph or state the solution.
Example 1 — Subtract
Solve \(x+3<7\).
Solution
Subtract \(3\).
| \(x\) | < | \(4\) |
Example 2 — Divide by positive
Solve \(2x\ge10\).
Solution
Divide by \(2\) (positive, no flip).
| \(x\) | \(\ge\) | \(5\) |
Example 3 — Divide by negative (flip)
Solve \(-3x>12\).
Solution
Divide by \(-3\) and flip the sign.
| \(x\) | < | \(-4\) |
Example 4 — Multiply by negative (flip)
Solve \(\dfrac{x}{-2}\le3\).
Solution
Multiply by \(-2\) and flip.
| \(x\) | \(\ge\) | \(-6\) |
Common pitfalls
Flip the sign for a negative multiplier or divisor — the most common error.
Don't flip for adding, subtracting, or a positive.
The solution is a range, not a single value.
Frequently asked questions
How is solving an inequality different from an equation?
The sign flips when you multiply or divide by a negative.
Solve \(-2x>6\).
\(x<-3\) — divide by \(-2\) and flip.
Does subtracting flip the sign?
No — only multiplying/dividing by a negative does.
What does the solution look like?
A range of values, often shown on a number line.
More in Single-variable inequalities