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It would make sense that as time increases, ln[A] should decrease as well. However, the rate constant, k, is always a positive number, therefore a negative sign must be put in front to make the slope of the line negative.
Remember that [A] is the concentration of the reactants, so, as Clarisse said, ln[A] will decrease over time. The fact that we're able to get k from the graph of ln[A] for a first order reaction is actually super neat.
The slope is negative because it will indicate which direction the gradient/slope will. You will see that when deriving the equation for the second order reaction, the negatives can cancel, making the gradient positive.
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