Linear Inequality Generator
Inequalities that solve exactly like equations until the moment they do not. Dividing both sides by a negative reverses the direction, and that single step is where most marks in the topic are lost. Rather than trusting its own rule, this generator verifies each answer by testing a value inside the claimed solution set and one outside it against the original inequality — if the direction were wrong, the question would not be shown.
What this generator does
Produces inequalities of the form ax + b < cx + d, using all four signs, and gives the solution as a range for x. The working collects terms, divides, and says explicitly whether the sign turned round and why.
How to use this tool
- Choose how many inequalities you want and at what level.
- Solve each for x, watching the sign of the coefficient you divide by.
- Reveal the answers to check both the boundary and the direction.
- Show the working to see where the sign turns round and why.
Understanding the controls
- How many questions
- Between 1 and 20. Twelve gives a fair chance of meeting several sign flips in one set.
- Level
- Gentle keeps the unknown on the left only. Standard and stretch put x on both sides, which is what makes the coefficient you finally divide by unpredictable in sign.
- Seed
- Reproduces the same inequalities from a word you pick.
- Show the answers / working
- The solution range, or the collection, division and flip step by step.
Common use cases
- Drilling the divide-by-a-negative rule until it is automatic
- Practice with the unknown on both sides at the harder levels
- Questions that keep the unknown on one side while the notation is still new
- Checking the four inequality signs are all read correctly
- Handing a class the identical set of inequalities from a seed
How this generator works
Coefficients are drawn so the x terms do not cancel, and the boundary is computed as an exact fraction. The check makes the two sides exactly equal at the boundary, then tests a point one unit inside the stated solution set and one unit outside against the original inequality. Both tests must agree with the direction shown.
Randomness and fairness
The coefficients and which of the four signs is used are random; whether the sign flips is determined by those coefficients, and is verified. Seeded sets reproduce exactly and are therefore explicitly not cryptographically secure. Unseeded draws come from the browser's cryptographically secure generator.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Only linear inequalities in one unknown. Quadratic inequalities and pairs of simultaneous inequalities are not covered.
- Boundaries can be fractions, which is realistic but means answers are not always tidy.
- Solutions are given algebraically. No number line is drawn.
- Double inequalities of the form a < x < b are not among the forms generated.
- Nothing is retained between visits; seed a set you want to reissue.
Privacy and your data
Inequalities are generated and verified in your browser. Your seed and level are not transmitted and are not remembered.
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