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In a dual-thrust solid fuel rocket motor, the propellant mass is composed of two different types or densities of fuel. In the case of a tandem dual-thrust motor, the fuel closest to the rocket nozzle burns fast and the fuel further into the motor's body burns slower. This has the effect of giving the rocket a lot of thrust initially, accelerating it up to high speed rapidly, and then once all the fast-burning propellant has burnt, the slow-burning propellant begins delivering a much lower level of thrust. The first phase of acceleration is called "boost" and the second phase "sustain". Not all dual-thrust motors are in a tandem arrangement but non-tandem motors function much the same; they just have a different physical layout of fuel, for example they might burn from the inside to the outside (core burning), rather than from the end in (end burning).
The advantage of dual-thrust motors is that, if the fuel were all of the fast-burning kind, the rocket would accelerate up to a higher speed initially but because air resistance increases quadratically with speed, the rocket would slow down very rapidly. This would give a higher peak speed but a lower average speed. Instead, the boost phase accelerates the rocket up to a high enough speed (high enough to propel the rocket to its destination fast, but not so high as to cause too much air resistance), and then the sustain stage allows the rocket to maintain this high speed for a while until it burns out, after which time it is able to coast and slowly lose speed.
Dual-thrust motors are most prevalent in rockets which are atmosphere-bound since they have to deal with air resistance over most of their flight. It is similar in concept to that of multi-stage rockets but much simpler to design and build since there is no requirement to detach stages, have separate components, etc.