Method guideAlgebra IIntermediate10 min read

Compound inequalities: and, or, and the sign flip

Solve each condition, reverse the inequality only when multiplying or dividing by a negative, then combine by intersection or union.

a<x<b\qquad x<a\ \text{or}\ x>b

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Updated July 13, 2026

LogicStart here

And means both conditions must hold, so keep the overlap. Or means either condition may hold, so keep the combined regions.

The inequality sign records order

Adding or subtracting the same amount preserves order. Multiplying or dividing by a positive also preserves it.

A negative scale reverses the number line, so the inequality direction must reverse. This is not a memorized exception; it follows from order.

-2x<8\Rightarrow x>-4

A double inequality such as 2 < x ≤ 7 asks for values satisfying both a lower and upper bound. Solve all three parts with the same operation.

Interval notation uses parentheses for excluded endpoints and brackets for included endpoints.

2<x\le7\Longleftrightarrow(2,7]

An or statement keeps values from either branch. The solution may be two separate rays rather than one interval.

Absolute-value inequalities often produce these patterns: less than gives an inside interval, while greater than gives outside rays.

|x|>3\Longleftrightarrow x<-3\ \text{or}\ x>3

Worked exampleSolve all three parts

-6\le2x<8

Subtract one from every part.

-3\le x<4

Divide every part by positive two.

[-3,4)

Translate included and excluded endpoints into interval notation.

Result

\boxed{-3\le x<4}

Watch for

Keep