Compound inequality solver

A compound inequality joins two statements about the same unknown with and or or. The and case is the intersection of the two solution sets: every accepted x must satisfy both linear comparisons at once. The or case is the union: x is allowed to satisfy either comparison, including both when they overlap. Students meet both forms in algebra one when they write pairs such as two x plus three less than seven and x minus one greater than zero, or when they write a split such as x less than two or x greater than five. This compound inequality solver isolates x in each linear piece ax plus b compared with c, flips the comparison whenever a coefficient is negative, then intersects or unites the resulting rays and returns the combined answer in interval notation. Empty intersections become the empty-set symbol; overlapping or adjacent or-rays that cover every real become the whole line; disjoint or-rays stay a two-piece union.

What a compound inequality is and how and versus or change the set

A compound inequality is not a single comparison and not a notation rewrite of an already-isolated x. It is two linear statements about the same variable, joined by a logical connector that decides whether both must hold or whether either is enough. In classroom language, and means the overlap of the two solution sets: if the first piece says x is greater than negative two and the second says x is at most four, the combined answer is the open-closed window from negative two to four.

Or means the coverage of either set: if the first piece says x is less than two and the second says x is greater than five, the combined answer is the disjoint union of a left ray and a right ray. Those two pictures look nothing alike on a number line, which is why a dedicated compound inequality solver exists instead of asking a student to run a single linear solver twice and combine the rays by hand.

Each piece here has the genuine linear form a x plus b compared with c, so a negative slope still reverses the inequality when both sides are divided—the same flip rule taught in algebra one. After isolation, every piece is a half-line. Intersecting two half-lines can yield a bounded interval, a tighter ray, a singleton when both ends close on the same number, or the empty set when the rays miss each other.

Input contract, the flip rule, and how the two rays are combined

The first piece uses a1, b1, c1, and op1 for a1 x plus b1 compared with c1; the second piece uses a2, b2, c2, and op2. Each coefficient must be a finite number, and a1 and a2 must be non-zero because a zero slope would make that piece independent of x and no longer a linear inequality in the unknown.

Operators may be written as symbols less than, less than or equal, greater than, or greater than or equal, or as the short codes lt, le, gt, and ge, including the unicode less-or-equal and greater-or-equal glyphs. Optional precision from zero through fifteen decimal places rounds coefficients and bounds before the set algebra so goldens stay stable across runtimes; the default of ten matches other tier-A algebra helpers. After subtracting the constant, both sides of each piece are divided by the slope.

A positive slope keeps the original operator. A negative slope reverses it, and the response sets flipped1 or flipped2 so a teaching layer can highlight that step. The engine then intersects the two rays when join is and, or unites them when join is or.

Where a compound inequality solver belongs in study, teaching, and code

Students meet compound inequalities when a word problem demands a range that satisfies two constraints at once, such as a budget that must stay below one limit and above another, and when a split condition says a reading is either too low or too high. Teachers can generate answer keys that show the isolated rays, the flip on a negative coefficient, and the final interval without trusting opaque calculator apps that forget the connector or print scientific notation.

Contest practice often needs a fast offline check that an empty intersection is truly empty and that an or of overlapping rays is the whole line rather than a sloppy union of two intervals that actually cover everything. This page is a different search intent from a single linear inequality solver, which isolates one comparison and stops, and from an interval-notation converter, which starts from an already-isolated x compared with a bound or from a chain that is already written as lower related to x related to upper.

Here the work is the solve-then-combine step that those tools leave to the human. On this page the calculation runs entirely in your browser.