If you do actually mean 'Brine tank drain hose', then that is an overflow drain hose for the brine tank which will not normally flow any water/brine but is only utilized if there should be a malfunction which results in the brine tank becoming overfilled with water, thereby preventing water from pouring out over the open edge of the brine tank. Any overflow will drain by gravity and so the line cannot rise above the drain fitting whatsoever but must discharge to a drain located lower than the brine tank's drain connection.
If you are actually referring to the control valve's drain line, then that will be utilized during each regeneration cycle. Drain flow will be pressurized by the water supply pressure, so depending on the supply pressure, the line maybe routed some distance vertically above the height of the softener, and also extended horizontally.
Where is the location of drain that the drain line is to drain into?
If the drain line is to discharge a substantial distance below the elevation of the softener, then a vacuum breaker would be recommended, located at the highest point in the drain line. A vacuum breaker will ensure the softener will not undergo a vacuum created by negative pressure by the water in the drain line that is being discharged below. The softener's media tank is designed for positive pressure, but if subjected to negative pressure, the tank may become crushed inward, which can occur under certain conditions.
While the pressurized drain line does not require any slope to the actual plumbing drain, when a vacuum breaker is utilized, then sloping the section of line after the VB is recommended, as negative pressure after the VB will cause the VB to admit air , thereby eliminating the vacuum condition, while allowing that section of drain line to drain by gravity.
The diagram directly below, may not pertain exactly to your application, but provides a general overview.
The photo of the crushed tank below, shows what will often occur when a media tank is subjected to a vacuum condition.