When a bracket sits in the way, the number of solutions cannot be read until it is multiplied out. These free printable worksheets ask how many solutions — after expanding first.
2(x + 3) = 2x + 6 looks like it might have one solution until the bracket is opened: 2x + 6 = 2x + 6, the same on both sides, so every number works. 2(x + 3) = 2x + 5 expands to 2x + 6 = 2x + 5, which is never true, so none does. The expanding is the step that reveals the answer, and it is what this page is about.
Why this sheet works
None of these can be answered by reading the two sides as printed — that is the point of splitting them off from the direct page. A child has to distribute the bracket, then collect the x terms and see what is left, the same three outcomes as before: one solution, none, or infinitely many. It is the harder half of the same standard, and the one a test is most likely to use to tell real understanding from pattern-matching.
How this one works
One question from this sheet, worked through a step at a time.
4(2x − 3) = 8x − 12
4(2x − 3) = 8x − 12
4(2x − 3) = 8x − 12
4(2x − 3) = 8x − 12
x is gone
−12 = −12— always 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 −12 = −12, which is true whatever x is. So every number works — infinitely many solutions.
