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 function made of the following operations is piecewise continuous:

Addition, subtraction, multiplication, and division;
Taking opposites, reciprocals, and absolute values;
Piecewise defined expressions whenever the conditions are given by intervals;
Raising to powers whenever the base is always positive;
Raising to powers whenever the exponent is always a whole number;
Extracting roots whenever the radicand is always positive;
Extracting roots whenever the index is always a whole number;
Taking logarithms (as long as they're defined as real numbers);
Applying any of the trigonometric or inverse trigonometric operations from Chapters 7 and 8 that you might learn about in Trigonometry.

Because of the fine print, there are potential exceptions here; if I want to solve (−2)^x < 1, for example, then I can do it, but not by this method directly. But of course, that's exactly the sort of exponential function that we refused to consider in Section 6.3!

Here is the method:

Turn the inquality into an equation.
Solve the equation (this is generally the hard part).
Also find when the expressions in the original inequality are undefined.
Finally, find all of the endpoints in the intervals of a piecewise defined function.
Using the numbers found in the previous two setps, pick one number between each pair of consecutive numbers, as well as one number on either side, as long as the function is defined there.
For each of the numbers found in the previous steps, check whether the inequality is true or false there.
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, so now you can read off the answer.

For rational functions, this method is in the textbook, but it still applies to other expressions, such as those inolving roots (of constant index) or logarithms.

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This web page was written in 2015 by Toby Bartels, last edited on 2015 December 8. Toby reserves no legal rights to it.

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