Solve a chained inequality lower < or ≤ ax + b < or ≤ upper. Beside solution interval, the output shows the operation used to obtain it.
−2<3x+1≤10 becomes −3<3x≤9 and then −1<x≤3.
A chained inequality requires the middle expression to satisfy both a lower and an upper condition simultaneously.
Check Compound Inequality from Lower bound, then verify Lower endpoint. Estimate the Compound Inequality Solution interval before computing it again. If Coefficient a changes during Compound Inequality, keep Lower endpoint fixed. That Compound Inequality comparison shows whether Solution interval moves as expected.
The lower bound must not exceed the upper bound. Dividing all parts by a negative coefficient reverses and reorders the comparisons.
Subtract b from all three parts, divide by a, reverse symbols when a is negative, and express the ordered result in interval notation. To continue from Compound Inequality, try symmetric distance bound.
Ranges, tolerances, domain restrictions, and acceptable operating bands commonly use compound inequalities.
Parentheses mark excluded endpoints and brackets mark included endpoints. Infinity always receives a parenthesis because it is not a reachable number. Translating the solved chain into interval notation should preserve both comparison directions and each endpoint’s strict or inclusive status.
Substitute the Compound Inequality solution into Lower bound. This Compound Inequality check rejects false Compound Inequality branches and forbidden denominators.
Choose an easy Lower bound value before running Compound Inequality. Predict Lower endpoint, then compare it with the Compound Inequality output.
Try a one-step sensitivity check on Compound Inequality: nudge Lower bound, freeze Lower endpoint, and predict the direction of Solution interval. The recalculated Solution interval should support that prediction unless the Compound Inequality formula crosses a boundary or changes branch.
Check the direction of Solution interval by changing Lower bound slightly. Hold Lower endpoint steady during this Compound Inequality trial. The new Solution interval should move as the Compound Inequality relationship predicts unless the calculation crosses a stated boundary.
Both comparisons must hold.
When the endpoint is included.
Both comparison directions reverse.
Yes, when transformed bounds conflict.
The constant statement is either always true or false.