Why did the cotton gin lead to slaves becoming more valuable than without the cotton gin?
The cotton gin sped up how quickly cotton seeds could be separated, but did not speed up the process of picking cotton. Slavery increased to pick cotton at a quicker pace.
The Southern and Middle colonies (and later States) on the Eastern Seaboard were built to become wealthy. Lacking the climate for tropical goods or furs, they relied instead on cash crops: tobacco, cotton, indigo, etc. Each of these crops required large plantations where they would be grown, and tons of labor.
During the industrial Revolution, cotton became the cash crop of choice in the South. There are three main steps to making cotton a trade good: it must be picked, processed, and then made into cloth. First, the creation of factories in both New England and Britain made the process of creating cloth much cheaper. Factories with underpaid workers could produce cloth from fibers much faster than before. This increased the demand for processed cotton.
Second, Eli Whitney invented the cotton gin to speed up the process of separating seeds from fibers after the cotton was picked. This sped up the processing time for cotton, increasing the demand for picked cotton.
To supply more cotton, plantation owners in the South vastly expanded the areas used for cotton (stretching the plantation lifestyle into newer territory in Georgia, Alabama and Mississippi). More and more slaves were required to pick the cotton on these plantations.