Linear Inequalities Calculator
This linear inequalities calculator solves ax + b against cx + d for any of the four symbols, and then does something no other one bothers with: it tests its own answer at three places in the inequality you actually typed.
Type 2x + 5 ≤ 11 and you get x ≤ 3. Type −3x + 4 > 10 and you get x < −2, with the point at which the symbol turned round marked out.
Solving One Is Almost Boring
Everything you already do to an equation, you do to an inequality. Collect the x terms on the left, the numbers on the right, divide by whatever is in front of the x.
2x + 5 ≤ 11 → 2x ≤ 6 → x ≤ 3
Nothing there is new. Which is precisely why the topic catches people out — the method feels familiar enough that attention wanders, and there is exactly one place where it must not.
The Moment the Symbol Turns
Divide or multiply both sides by a negative number and the symbol reverses.
−3x + 4 > 10 → −3x > 6 → ÷ (−3) → x < −2
Every textbook tells you that. Almost none tells you why, and the why is short enough to fit in one line:
2 < 5 but −2 > −5
Look at what happened. Two is smaller than five; negate both and two becomes the bigger one. Multiplying by a negative reverses the order of every pair of numbers on the line — and an inequality is nothing more than a claim about which of two numbers comes first. When the line turns round, the claim about it has to turn with it.
Once you have seen it that way it stops being a rule to remember. It is simply what negatives do, and inequalities inherit it.
The Other Flip, Which Nobody Mentions
Here is the one that costs marks quietly. Suppose you finish a question and your working leaves you with the x on the right:
6 > 2x → 3 > x
Now you tidy it into the usual shape, x first. It is very tempting to slide the x across and leave the symbol alone. Do not. 3 > x and x < 3 are the same statement; x > 3 is its opposite.
No negative number appeared anywhere in that step. Nobody warned you, because the warning you were given was about dividing. Swapping the sides is the second trigger, and it is the more dangerous of the two precisely because it arrives at the end — someone who knows the negative rule perfectly can still reverse their own answer on the last line, while tidying up.
This calculator prints both forms side by side for exactly that reason.
Checking an Answer That Is a Whole Range
With an equation, checking is easy: substitute and watch it balance. With an inequality there is no single number to substitute, so students tend to skip the check altogether.
Do it the other way round. Pick a number that should work, and one that should not.
For x < −2, try −3 and try −1, in the original inequality:
| x = −2 | the boundary | 10 > 10 is false, so the end is out |
| x = −3 | inside | 13 > 10 is true ✓ |
| x = −1 | outside | 7 > 10 is false ✓ |
If the number inside fails, or the one outside passes, your symbol is the wrong way round. Two substitutions catch it every single time, and they take about ten seconds. The working on this page runs all three on every answer, in your original inequality rather than the tidied one — because a mistake made while tidying will happily survive a check against the tidy version.
Open End or Closed End
A strict symbol (< or >) leaves the boundary out. A non-strict one (≤ or ≥) takes it in. On a number line that is a hollow circle against a filled one; in interval notation it is a round bracket against a square one.
| x < 3 | (−∞, 3) | hollow circle at 3 |
| x ≤ 3 | (−∞, 3] | filled circle at 3 |
| x > 2 | (2, ∞) | hollow circle at 2 |
| x ≥ 2 | [2, ∞) | filled circle at 2 |
Infinity never gets a square bracket. It is not a number you can reach, so there is nothing to include.
One character decides right from wrong here, which is why the boundary is worth testing on its own. Put it back into the original and see whether the statement holds. If it does, the end belongs to the answer.
When the x Disappears
Sometimes both sides climb at the same rate and the x terms cancel, taking the question with them.
2x + 1 < 2x + 5 collapses to 1 < 5. That is true, and it has nothing to do with x — so every number is a solution.
2x + 5 < 2x + 1 collapses to 5 < 1. False, always. No number works.
Both of these are complete answers. Students who reach this point often assume they have gone wrong and stop, leaving the page blank, which scores nothing. Write down which of the two it is and say why in a few words. On paper, a blank space and a wrong answer look identical to the person marking it.
Using the Calculator
The four boxes are laid out exactly as the inequality is written — a and b on the left, the symbol between, c and d on the right. If your right-hand side is only a number, leave c alone; it counts as zero. Fractions like 3/4 work, decimals work, and the answer stays exact either way: 3x + 1 < 2 comes back as x < 1/3 rather than 0.333.
Open the working and you get the collecting step, the division with the turn called out at the moment it happens, both ways of writing the answer, the number line with the correct kind of end, and the three tests. If you are also working with absolute value equations, the same habit of testing rather than trusting applies there for a different reason.
Linear Inequalities FAQ
How do you solve a linear inequality?
When do you flip the inequality sign?
Why does the inequality sign flip?
What is interval notation for an inequality?
Can a linear inequality have no solution?
How do you check the answer to an inequality?
Collect, divide, and watch the symbol. Then test it — one number in, one number out.
