it is stated in your textbook that we could chose to calculate the alkalinity of a water sample either exactly, by including all typical terms in the alkalinity equation, namely the carbonate, bicarbonate, hydroxyl, and hydrogen ions, or that we could calculate the approximate alkalinity of a sample by including only the two carbonate species and ignoring the hydroxyl and hydrogen ions.
Such a distinction between approximate and exact alkalinity, based on separation between carbonate and non-carbonate ions, is incorrect. Demonstrate the authors' reasoning error by considering the importance of each ion included in the alkalinity equation as a function of pH. Clearly, ions expected to contribute significantly to alkalinity, cannot be excluded when estimating the approximate alkalinity, while ions with minor contribution could be safely ignored.
HINT: Simplify your analysis by separating the entire pH range into three regions: low pH, around neutral, and high pH. Show that in none of these three pH regions, the distinction between approximate and exact alkalinity could be correctly made based simply on separation between carbonate and non-carbonate species and that, therefore, the authors' statement is wrong, regardless of the pH of the water.