- 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 a constant;
- Extracting roots whenever the radicand is always positive;
- Extracting roots whenever the index is a constant;
- Taking logarithms (as long as they're defined as real numbers);
- Bonus: Applying any of the trigonometric or inverse trigonometric operations from Chapters 7 and 8 that you might learn about in Trigonometry.

Here is the method:

- Turn the inquality into an equation and solve it.
- Besides these solutions, also find when the expressions in the original inequality are undefined.
- Finally, if you have a piecewise-defined function in the problem, find all of the endpoints in the intervals of the pieces' conditions.
- Using the numbers found in the Steps 1–3, 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 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.

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This web page was written in 2015 and 2016 by Toby Bartels, last edited on 2016 May 25. Toby reserves no legal rights to it.

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