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a very common problem in mathematics is to determine the sign of an expression involving a variable and how the sign of the expression varies with X lets consider a very simple example the expression X minus 5 the goal is to determine the sign of this expression for different values of X this is quite straightforward we can start by noting that X minus 5 is equal to 0 when X is equal to 5 since X minus 5 will increase with X we also see that X minus 5 is positive when X is greater than 5 now we see that X minus 5 is negative when X is less than 5 a sign diagram for the expression X minus 5 will then look as follows we have the real line with 0 here we put the expression under the real line to the left like this then we identify all the points on the real line when the expression takes the value 0 in this case it is only the number 5 we do a line and we put a 0 here then we put minus signs here since the expression takes a negative value when X is less than 5 finally we put plus signs
