How to Use the System of Linear Inequalities Solver
Enter the coefficients and operators for two linear inequalities in the form a1x+b1[operator]0 and a2x+b2[operator]0, and this calculator finds each individual solution plus the common solution interval that satisfies both. It correctly handles the sign flip that happens when a coefficient is negative — a common source of manual errors.
When solving a linear inequality for x, dividing both sides by a positive number keeps the inequality sign unchanged, but dividing by a negative number flips it. For example, solving -2x+4>0 gives -2x>-4, and dividing both sides by -2 gives x<2 — the sign flips from '>' to '<'. This calculator applies that rule automatically for each inequality.
Once both solutions are found, the calculator compares the lower-bound conditions (≥ or >) and takes the larger value, then compares the upper-bound conditions (≤ or <) and takes the smaller value, to find the overlap. If the two intervals don't overlap at all, there's no common solution (empty set); if they meet at exactly one point, that point is the unique solution. Useful for checking algebra homework or verifying inequality logic in code.
Frequently Asked Questions
Dividing both sides of an inequality by a negative number flips the inequality sign. For example, solving -2x+4>0 for x gives x<2 — the original '>' becomes '<'.
If the two intervals share no overlap at all, there is no common solution. In that case the system's solution is shown as 'no solution (empty set)'.