The ultimate inequality challenge
JavaScript (ES6),  58 45  42 bytes
Saved 3 bytes thanks to @Neil
Expects the Unicode code point for \$c\$. Returns \$0\$ or \$1\$.
(a,c,b)=&gt;'14353426'[c%61%9]&gt;&gt;(a&gt;b?2:b&gt;a)&amp;1

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How?
Each comparison character is assigned a 3-bit mask describing if it should be truthy for a &gt; b, a &lt; b or a == b.
 char. | code | meaning                           | &gt; | &lt; | = | mask
-------+------+-----------------------------------+---+---+---+------
   &lt;   |   60 | Less than                         | 0 | 1 | 0 |  2
   =   |   61 | Equal to                          | 0 | 0 | 1 |  1
   &gt;   |   62 | Greater than                      | 1 | 0 | 0 |  4
   ≠   | 8800 | Not equal to                      | 1 | 1 | 0 |  6
   ≤   | 8804 | Less than or equal to             | 0 | 1 | 1 |  3
   ≥   | 8805 | Greater than or equal to          | 1 | 0 | 1 |  5
   ≮   | 8814 | Not less than                     | 1 | 0 | 1 |  5
   ≯   | 8815 | Not greater than                  | 0 | 1 | 1 |  3
   ≰   | 8816 | Neither less than nor equal to    | 1 | 0 | 0 |  4
   ≱   | 8817 | Neither greater than nor equal to | 0 | 1 | 0 |  2

We store these masks in a 8-character lookup string whose index is computed by applying two consecutive modulos to the code point:
 code | mod 61 | mod 9 | mask
------+--------+-------+------
   60 |   60   |   6   |  2
   61 |    0   |   0   |  1
   62 |    1   |   1   |  4
 8800 |   16   |   7   |  6
 8804 |   20   |   2   |  3
 8805 |   21   |   3   |  5
 8814 |   30   |   3   |  5
 8815 |   31   |   4   |  3
 8816 |   32   |   5   |  4
 8817 |   33   |   6   |  2