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Is it not enough to validate a transaction simply by confirming that the sender actually has X amount to send? What benefit is there to using weights?

And even if that transaction is already confirmed by another, what is preventing that same sender from sending more than Old total - X. Couldn't the double-spend transaction just attach to a different part of the tangle such that there is no path from the double-spend to the first transaction?

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Weighted transactions

It's all about the consensus on the IOTA network. If the transanctions didn't have weight (unweighted random walk) lazy tips would be approved as any other without being penalized.

With weighted random walk the probability of a lazy tip being picked by the walker decreases since it has a low cumulative weight. You can get more details on part 3 of the The Tangle: an Illustrated Introduction.

More on that, the weighted random walk approach provides protection against “parasite chain” attacks as described on section 4.1 of the whitepaper.

Double-spending

Couldn't the double-spend transaction just attach to a different part of the tangle such that there is no path from the double-spend to the first transaction?

You can do it, but the tangle will keep growing and the nodes will have to decide which one they will approve because the transactions are conflicting, the branch who grows faster is the winner. This is the adaptation period, that's why there is the concept called confirmation confidence where you wait until you are sure about what branch won.

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