Inequality Chains

• Same direction: $A > B \ge C$ gives $A > C$. All the signs point one way, so the relation holds.
• Strict or not: if at least one sign on the way is strict ($>$ or $<$), the result is strict. If every sign allows equality ($\ge$, $=$), the result is $\ge$ or $\le$.
• Equal signs never change the direction; they can be passed freely.
• Opposite directions: $A > B < C$ gives no relation between A and C.
• Statements split by commas or semicolons can be joined through a shared letter: $T \ge A$ and $A \le D$ do not join into one direction, but $C < A$ and $A \le T$ do.
• Fast route: write just the stretch of the chain between the two letters of a conclusion, and look at the signs alone.

• Either-or pair: the two conclusions name the same two letters, neither is definite on its own, and together they cover every case.
• When the statement gives $X \ge Y$: the pair $X > Y$ and $X = Y$ is an either-or pair.
• When the statement gives no relation at all: the pair $X \ge Y$ and $X < Y$ is an either-or pair, since one of them must hold.
• A pair that leaves a case out ($X > Y$ and $X < Y$, missing $X = Y$) is not an either-or pair.
• Fast route: only check for either-or after both conclusions have failed on their own.

1. Copy the table of meanings: for example @ for $\ge$, # for $<$, % for $=$. 2. Rewrite every statement with ordinary signs, and join the statements through their shared letters. 3. Rewrite each conclusion the same way and test it with the rules for reading a chain.

• Fast route: decode the conclusions first; often one of them already contradicts a statement.