How can someone determine if a support or resistance line has been successfully broken or not?

Support and resistant line for me it is really broke when it has been retest. It mean after price when thru the S/R line, it go back S line now become R, R become S. It is then I consider it were broken. There are cases when retest does not happen, which are: R/S are not strong so price easily go thru. In this case you should bot consider that R/S for your trade plan. Price momentum are too strong base on some event, in that case should not trade neither

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