How many different isosceles triangles have integer side lengths and perimeter 23?

Let  x be the equal sides of the  triangle...let y be the remaining side

One way of solving this is with a graph...

We have these constraints  

2x + y = 23

And by the Triangle Inequality, we have

2x > y

x + y > x   ⇒  y > 0

Look at the graph of the feasible region here :  https://www.desmos.com/calculator/kyzw7wa8v5

It shows that there are six such triangles