2x+1=2x+1 is an example of an identity that has an infinite amount of solutions

Yes. 2x+1 is equals to 2x+1 which means that it has an infinite amount of solutions.
The only way it doesn’t have an infinite amount of solutions is if the result is different.
For example, 4x -4= 8
The -4 goes on the other side which turns it into +4.
4x= 12
Divide by 4 on both sides.
This gives you x= 3.
This is an example of only one solution.

2x+3= x+3
2x+3 is not equal to x+3.
So this is a no solution.

3x+6+2x= 5x+6
Add the like terms together.
3x+2x+6= 5x+6
5x+6 = 5x+6
This statement is true, 5x+6 is equal to 5x+6.
So this is an infinite amount of solutions.

We need more info then that