Compound inequalities: and, or, and the sign flip

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.

And means intersection

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.

Or means union

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.

Worked example

Common mistakes

• Flipping the sign after adding a negative instead of multiplying by one.
• Keeping a union for an and statement.
• Using a bracket at a strict endpoint.

Keep these ideas

• Only negative multiplication or division flips order.
• And keeps overlap.
• Or keeps the union.