She spoke loudly ____ all might hear.

'in order that': is the right answer because 'in order that' means 'for the reason,' which indicates the sentence's goal. If this option is selected, it signifies that she talked loudly enough for everyone there to hear her.
'on condition that': is an incorrect option because it provides a condition under which the action should occur. If this option is selected, the meaning of the statement will be changed to 'if all may hear her,' else she will not speak, changing the original sentence meaning.
'even if': is an invalid option since it expresses speculation, which refers to a hypothetical or imaginary circumstance that does not fit with the phrase because the activity has already taken place.
'Provided that': is an incorrect option because it refers to the condition if fulfilled, which means 'the provision on something'. The preceding is not conditional, and if this choice is chosen, the sentence will no longer have its original meaning.