Compound Inequality Calculator
This compound inequality calculator solves any of the three ways one gets written — 3 < 2x + 1 < 7, x > 3 and x < 8, or x < −2 or x > 5 — and gives the answer in words, in interval notation, and on a number line.
You are not asked to pick a connector. If there is one written down it is read; if there is not, the chain form supplies its own.
What the Compound Inequality Calculator Has to Get Right
A compound inequality is two inequalities joined by and or by or. Solving each one is ordinary work. The whole topic sits in what happens next:
and → a number must satisfy both → keep the overlap → intersection
or → either one will do → keep everything either covers → union
It is worth checking you have those the right way round, because published pages get them backwards — including one of the largest maths sites on the web, on the page that ranks for this very keyword.
A Quick Way to Remember Which Is Which
Forget intersection and union for a second and think about what the word does to the answer.
and makes a number pass two tests, so it can only ever make the answer smaller. or lets it pass on one, so it can only ever make the answer bigger. If your and answer came out wider than either half you started with, something has gone wrong — and the same in reverse for or.
The Chain Form Has a Hidden AND
3 < 2x + 1 < 7 means 2x + 1 > 3 and 2x + 1 < 7. Nobody writes the word in, which is exactly why the form catches people out.
The middle expression has to satisfy the left condition and the right one at the same time, so every chain is an and. There is no chain that means or — if you find yourself wanting one, what you actually have is two separate inequalities that need the word written between them.
Solve Each Half Exactly as You Would Alone
Nothing new happens inside either half. Collect, divide, and deal with a negative divisor exactly as you always would — the same as if the other half were not there.
So −2x > 4 and x > −10 becomes x < −2 and x > −10, and only then does the connector get to do its job. That step belongs to the half it happened in; it never touches the joining word.
An AND Can End With Nothing
x > 5 and x < 2 has no answer. No number is above five and below two at the same time, so the solution set is empty.
Write that down as the answer rather than assuming you made a mistake. It is also the case a backwards definition hides: read and as a union and this looks like it has a solution, so the error survives unnoticed.
An OR Can Come in Two Pieces
x < −2 or x > 5 is not one range. It is two, with everything from −2 to 5 left out of the middle.
An and answer is never shaped like that — it is always a single stretch or nothing at all. So a gap in the middle tells you at a glance which connector you were working with, and a gap where you expected one range usually means the words got swapped.
Open End or Closed End
The endpoints are where compound answers go wrong, and the rule follows from the connector rather than from the symbols.
For and, an endpoint is in only if it is in both halves — so x > 3 and x ≤ 3 has no answer at all, even though the two look as if they meet. For or, it is in if it is in either, which is why x ≤ 3 or x > 3 covers every number while x < 3 or x > 3 leaves a single hole at 3.
Writing It in Interval Notation
A square bracket means the end is included, a round one means it is not, and infinity always gets a round one because you never reach it.
1 < x < 3 is (1, 3). −2 ≤ x ≤ 3 is [−2, 3]. A two-piece answer is joined with the union sign: (−∞, 3] ∪ (8, ∞). And an empty answer is written ∅.
Check the Ends, Not the Middle
The middle of a range is almost never where an answer goes wrong. Take each endpoint, put it back into both original inequalities, and combine the two results with the joining word by hand.
If an endpoint fails one half of an and but your answer includes it, the bracket is the wrong shape. This page runs that test on every endpoint before it shows you anything, along with a point a thousandth either side.
Using the Compound Inequality Calculator
Type the compound however you have it — with and, with or, or as a chain. You get the answer in words and in interval notation, each half solved on its own with any turned symbol pointed out, the connector applied with the reasoning shown, and a number line with both halves drawn above the answer so you can see the overlap being taken or the two being added together.
Compound Inequality Calculator FAQ
What is a compound inequality?
Does and mean intersection or union?
Can a compound inequality have no solution?
Why does my answer come in two pieces?
Does the chain form 3 < 2x + 1 < 7 mean and or or?
Solve each half, then let the word decide — and check the ends.