I want to highlight two different styles of solving problems. One is where you jump directly to the solution, and the other is where you follow a procedure to produce the solution.
Pushing and pulling. When you have a jigsaw puzzle, you’re done when you push the last piece into place. Trying to solve an equation for a variable by pushing might look like: x^2-1+3=5. x=0? 2 != 5. x=1? 3!=5. x=2? 6!=5. This might seem unproductive, but (using knowledge about x^2) we can see there’s a solution where 1<x<2. Pulling might look like: x^2-1+3=5. x^2+2=5. x^2 = 3. Or we could go meta*, solve ax^2+bx+c=d, then substitute the values in for the solutions.
The licence to substitute
The reason you can’t go from -x < y to x < -y: Going from 1 < 2 to 1 > 2 isn’t valid. (Replacing the abstract with the concrete.*)
Pushing and pulling. When you have a jigsaw puzzle, you’re done when you push the last piece into place. Trying to solve an equation for a variable by pushing might look like: x^2-1+3=5. x=0? 2 != 5. x=1? 3!=5. x=2? 6!=5. This might seem unproductive, but (using knowledge about x^2) we can see there’s a solution where 1<x<2. Pulling might look like: x^2-1+3=5. x^2+2=5. x^2 = 3. Or we could go meta*, solve ax^2+bx+c=d, then substitute the values in for the solutions.
The reason you can’t go from -x < y to x < -y: Going from 1 < 2 to 1 > 2 isn’t valid. (Replacing the abstract with the concrete.*)
*These are opposites.