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Is this an automatic or manual transmission?
If all of the forward gears work correctly and only the reverse is faulty, most likely the reverse band inside the transmission is faulty. To repair this the transmission will need to be pulled apart and inspected for other damage. The cost will most likely be the price of a rebuild.
Do all of the other gears work properly or is their a delay in forward shifting?
Is the overdrive light or the check engine light?
This question will be open for a while (at least for the next 2 days)
If the forward gears work correctly and their is no check engine light then you will need to have the transmission tore down to have the reverse band replaced and inspected for other damages. Most shops will do a complete rebuild as sometimes a complete rebuild kit may need to be ordered to get that part.