We see that (i) and (ii) have four terms each and hence there is no common term. Therefore, a combination is not possible for these as they fail in step I itself. For all others, there are three terms in all the pairs and common term is B, So, we apply step II to all of these.
Step II: (ii) B less than one term A, greater than another one C, combination possible.
Combined inequality, `A gt B gt C "or" C lt B lt A`
(iv) B less than (or equal to) C, and B less than A, combination not possible.
(v) B greater than (or equal to) both A and C, combination not possible.
(vi) B less than one term C, greater than another term A, combination possible.
Combined inequality: `A lt B lt C "or" C gt B gt A`
(vii) B less then (or equal to) both terms, combination not possible.
(viii) B greater than (or equal to) one term A and less than (or equal to) other term C, combination possible.
Combined inequality: `C ge B ge A "or" A le B le C`
(ix) B less than (or equal to) one term A and greater than (or equal to) other term C, Combination possible.
Combined inequality: `A ge B ge C "or" C le B le A`
(x) B less than one term C and greater than (or equal to) other term A, combination possible.
Combined inequality: `A le B lt C "or" C gt B ge A.`
Step III: (iv,(v),(vii) were rejected in step II.
Now, let us apply third golden rule to the combined inequality obtained in case of (iii), (vi), (viii), (ix) and (x).
(iii) Combined inequality: `A gt B gt C`
Conclusion: `A gt C`
(Because `.ge.` sign does not appear twice in the combined inequality, it does not appear in the conclusion.)
(vi) Combined inequality: `A le B le C`
Conclusion: `A lt C`
(Because `.le.` sign does not appear twice in the combined inequality, it does not appear in the conclusion.)
(viii) Combined inequality: `C ge B ge A`
Conclusion: `C ge A`
(because `.ge.` sign appears twice in the combined inequality, it does appear in the conclusion.)
(ix) Combined inequality: `A ge B ge C`
Conclusion: `A ge C`
(Because `.ge.` sign appears twice in the combined inequality, it does appear in the conclusion.)
(x) Combined inequality: `A le B lt C`
Conclusion: `A lt C`
(Because `.le.` wsign does not appear twice, it appears only once, it does not appear in the conclusion.)
Thus, our solution is:
`A lt B lt C`. Conclusion `A lt C`
