An object will move in spite of equal and opposite actions and reactions because for every action, there will always be an equal but opposite reaction.
The law of motion states that a forces always act in equal but opposite pairs.
In other word, the law asserts that for every action, there is an equal but opposite reaction.
Hence, that when one push an object, the object pushes back on you with a force equal in strength to the force you exerted.
In conclusion, an object will move in spite of equal and opposite actions and reactions because for every action, there will always be an equal but opposite reaction.
Read more about Newton's third law of motion
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