Solving inequalities (§5.4)

There is a very general technique for solving inequalities in one variable that applies to expressions built using pretty much all of the functions that we consider in this course. Specifically, it applies to all piecewise-continuous functions. Exactly what that means is generally explained in a Calculus course, but I can already tell you what examples we have of these: any real-valued function made of the following operations is piecewise-continuous:

Addition, subtraction, multiplication, and division;
Taking opposites, reciprocals, and absolute values;
Raising to powers whenever the exponent is a constant;
Extracting roots whenever the index is a constant (which is usually always the case when people write things with roots);
Partially-defined or piecewise-defined expressions whenever the conditions are given by intervals;
Raising to powers whenever the base is always positive;
Extracting roots whenever the radicand is always positive;
Taking logarithms;
Bonus: applying any of the trigonometric or inverse trigonometric operations from Chapters 7 and 8 that you might learn about in Trigonometry.

This is a long list, but there are potential exceptions here: if you want to solve (−2)^x < 1, for example, then it can be done, but not directly by this method; the problem is that the base is not positive and the exponent is not constant.

Here is the method:

Turn the inquality into an equation and solve it. (If you get an entire interval of solutions, then you can just keep the endpoints here.)
Besides these solutions, also find when the expressions in the original inequality are undefined. (Again, if you get an entire interval, then you can just keep the endpoints.)
Finally, if you have a partially-defined or piecewise-defined function in the problem, then find all of the endpoints in the intervals of the pieces' conditions.
Using the list of numbers found in Steps 1–3, pick one number between each pair of consecutive numbers in the list, as well as one number on each side (positive and negative) beyond the list, as long as the function is defined there.
For each of the numbers found in Steps 1–4, check whether the inequality is true or false there.
Now you can read off the answer, letting each number found in Step 4 speak for all of the numbers in the open interval from which it was chosen.

This works because the only way for the inequality to shift from true to false is by going through a place where the equation is true or undefined or by switching from one piece to another in piecewise-defined examples. This method is in the textbook for rational functions (which are normally written using only the first three items in the list of operations and always can be written using only the first item in the list), but it still applies to other expressions involving any or all of the operations listed.

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This web page was written in 2015 and 2017 by Toby Bartels, last edited on 2018 May 20. Toby reserves no legal rights to it.

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