If You Replace The Equal Sign Of An Equation With An Inequality Sign, Is There Ever A Time When The Same Value Will Be A Solution To Both The Equation And The Inequality?

The same value will be a solution to the equality and the inequality if the inequality is of the kind
  - less than or equal, or
  - greater than or equal.
A value that is "equal" will never be a solution to an inequality requiring the solution to be "not equal."