Direct answer
Solve a linear inequality with equality-preserving steps, but reverse its comparison sign when multiplying or dividing by a negative number; the result describes a set rather than usually one value.
What this calculation tells you
An inequality describes values ordered above, below, or between boundaries. Its solution contains every value that makes the comparison true.
Open endpoints exclude equality; closed endpoints include it. Constant inequalities may be true for every real value or none.
Where it is used
Capacity and planning
Represent upper and lower operating limits.
Eligibility
Express threshold rules with correct inclusion.
Measurement
Model tolerance bands and one-sided limits.
Algebra and graphing
Connect symbolic comparisons with number-line regions.
When this guide helps
- Solving ax+b≤0.
- Deciding whether a threshold value is included.
- Writing a number-line interval.
- Checking a negative-coefficient sign reversal.
Preserve order
Adding the same quantity preserves order. Multiplying by a negative reflects the number line and therefore reverses the relation.
Treat the answer as a set
Use interval notation, a number line, or a verbal boundary. One test point can help confirm the shaded direction.
Keep threshold definitions exact
Words such as below, at most, more than, and at least map to different strict or inclusive symbols. Do not infer inclusion from a rounded display.
Common mistakes
Typical errors are forgetting the negative sign reversal and reporting only the boundary value.
- Translate threshold words.
- Use a test value.
- State inclusion explicitly.
Worked case: isolate a variable
Solve 3x - 6 < 9.
Add 6 to get 3x < 15, then divide by positive 3 to get x < 5.
The solution set contains all real x below 5, excluding 5.
A test value such as x=4 satisfies the original inequality.
Reproduce this worked caseOpen Inequality Calculator
Worked case: dividing by a negative flips direction
Solve -2x > 8.
Divide both sides by -2 and reverse the sign, giving x < -4.
The solution is x < -4.
Keeping > would fail: x=0 would appear allowed even though 0 > 8 is false in the original expression.
Reproduce this worked caseOpen Inequality Calculator
Compare inequalities cases before generalising
Strict inequalities exclude the endpoint; ≤ and ≥ include it. Domain restrictions and compound conditions can split the final solution into intervals.
| Inequality | Critical operation | Solution |
|---|---|---|
| 3x-6<9 | Divide by +3 | x<5 |
| -2x>8 | Divide by -2 and flip | x<-4 |
inequalities: calculation checklist
- Track strict versus inclusive sign
- Flip only for negative multiplication/division
- Find domain restrictions
- Test boundary and sample points
- Express interval endpoints correctly
Practical questions
Frequently asked questions
Why does dividing by a negative flip the sign?
Negative multiplication reverses the order of points on the real number line.
What is the difference between < and ≤?
The second includes equality at the boundary; the first excludes it.
Can an inequality have no solution?
Yes, or it can be true for every real value when its variable terms cancel.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.