With most of the HP printers that use 1 black cartridge, and 1 tri-color cartridge: The black cartridge is likely a pigment ink, and the tri-color cartridge is likely a dye ink.
I use these HP inks for the inkjet transfers for lights (I use
JPSS), and that works just fine.
As for the Ironall for darks, I did not have good results using the HP printer. Colors ran, and faded.
Printing the Ironall for darks with my Epson printer (which uses pigment inks) has produced very good results. The colors don't run, or fade. I have a few shirts that have been washed at least a dozen times, and the colors are still great.
I could not find your HP printer model listed, so I can't say for sure what kind of ink it uses, but if it takes one black, and one tri-color cartridge, then the color inks are dye, and not suitable for opaque transfers.
If your HP printer has 4 cartridges -- black, magenta, yellow, and cyan, then you have pigment inks, and that would be suitable for opaque transfers.
BTW -- I purchased my little Epson for I think $30 when Epson was running one of their re-furb specials. It came with the ink cartridges, which cost more than $30 to purchase seperately. So, if you're considering purchasing a printer that uses pigment inks, check the Epson site.