Interval Notations

JavaScript (ES6), 184 182 181 180 bytes

Takes input in currying syntax (a)(b). Returns an array of 4 strings.

a=>b=>[1,2,3,4].map(k=>'31,23 = 31,23 = {10x72}4{10x&&x72}45..6'.replace(/\d/g,(n,i)=>[(+n?k<3:k&1)?'<':'<=',a,b,'][)([[]('[(i*17^k*718)%9],'   = '.slice(k/2),a+k%2,b-(k<3)][n%7]))

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How?

For each row k with 1 ≤ k ≤ 4, we start with the following template:

"31,23 = 31,23 = {10x72}4{10x&&x72}45..6"

and replace each decimal digit n at position i according to the following table:

  n  | Replaced with           | Code
-----+-------------------------+------------------------------------------
 0,7 | comparison operator     | (+n ? k < 3 : k & 1) ? '<' : '<='
  1  | a                       | a
  2  | b                       | b
  3  | interval bound          | '][)([[]('[(i * 17 ^ k * 718) % 9]
  4  | a substring of '   = '  | '   = '.slice(k / 2)
  5  | either 'a' or 'a + 1'   | a + k % 2
  6  | either 'b' or 'b - 1'   | b - (k < 3)

Python 2, 225 203 195 bytes

a,b=input()
for d in 0,1:
 for m in 0,1:k=`a`+','+`b`;o='{'+`a`+'<'+m*'=';c='x<'+d*'='+`b`+'}'+'  '[m+d:];print' = '.join(['(['[m]+k+')]'[d],']['[m]+k+'[]'[d],o+c,o+'x&&'+c,`a+1-m`+'..'+`b-1+d`])

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Python 2, 187 bytes

t=a,b=input()
for j in 1,0:
 for i in 1,0:print"%%s%d,%d%%s = "%t*2%('[('[i],'])'[j],'[]'[i],']['[j])+"{%d<%sx%%s<%s%d}%s = "%(a,'='[i:],'='[j:],b,' '*(i+j))*2%('','&&x')+`a+i`+'..'+`b-j`

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