Algebra
Single-variable inequalities
Multi-step inequalities
20 practice questions
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Theory + worked examples
Theory
A multi-step inequality is solved in the same order as a multi-step equation:
- Distribute and combine like terms.
- Move variables to one side.
- Undo with inverse operations.
- Flip the sign if dividing/multiplying by a negative.
Only the negative-multiplier step flips the inequality.
The order for a multi-step inequality.
\(x>-3\) on a number line.
Same steps, one exception:
\[-2x<6\ \Rightarrow\ x>-3\]
Simplify each side first, then isolate.
How to solve multi-step inequalities
- Distribute across parentheses.
- Combine like terms.
- Collect variables on one side.
- Isolate, flipping the sign for a negative.
Example 1 — Two steps
Solve \(2x+1>9\).
Solution
Subtract, then divide.
| \(2x\) | \(>\) | \(8\) |
| \(x\) | \(>\) | \(4\) |
Example 2 — Distribute first
Solve \(3(x-1)\le12\).
Solution
Distribute, then solve.
| \(3x-3\) | \(\le\) | \(12\) |
| \(x\) | \(\le\) | \(5\) |
Example 3 — Negative coefficient (flip)
Solve \(-2x+5<11\).
Solution
Subtract \(5\), then divide by \(-2\) and flip.
| \(-2x\) | < | \(6\) |
| \(x\) | \(>\) | \(-3\) |
Example 4 — Variables both sides
Solve \(5x-2\ge2x+7\).
Solution
Collect variables.
| \(3x\) | \(\ge\) | \(9\) |
| \(x\) | \(\ge\) | \(3\) |
Common pitfalls
Flip only for a negative multiplier or divisor.
Distribute to every term.
Keep the sign correct through each step.
Frequently asked questions
How do you solve a multi-step inequality?
Like a multi-step equation, flipping the sign for a negative.
When does the sign flip?
Only when multiplying or dividing by a negative.
Solve \(-x+2<5\).
\(-x<3\), so \(x>-3\).
Do you distribute first?
Yes — clear parentheses before isolating.
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