Solutions after expanding, step by step

When a bracket sits in the way, the number of solutions cannot be read until it is multiplied out.

The question

How many solutions?

3(3x + 1) = 9x + 3

x is gone

3 = 3is true

infinitely many solutions

Do not start by hunting for x. Ask a smaller question first: what is left when the x terms are collected on one side?
The bracket has to come off before the two sides can be compared at all — multiply everything inside it by the number outside.
Both sides have the SAME number of x, so taking it off both sides removes x completely. That is not a mistake — it is the answer arriving early.
What is left is 3 = 3, which is true whatever x is. So every number works — infinitely many solutions.

How it works.

01
Do not start by hunting for x. Ask a smaller question first: what is left when the x terms are collected on one side?
02
The bracket has to come off before the two sides can be compared at all — multiply everything inside it by the number outside.
03
Both sides have the SAME number of x, so taking it off both sides removes x completely. That is not a mistake — it is the answer arriving early.
04
What is left is 3 = 3, which is true whatever x is. So every number works — infinitely many solutions.