Your transaction may not enter the mempool if the gas price is set too low. This means that the transaction is not broadcast at all and you have to use a higher gas price.
Not being broadcast is different from being failed.

Gas price in ethereum is something like fee rate in bitcoin.
If the gas price is set too low, the transaction may not be broadcast at all and even if it is broadcast, it will take a very long time to be confirmed. But it won't fail.

I know a transaction can fail due to different reasons. But a transaction isn't failed due to low gas price.