- 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 (see Section 5.4 of the textbook);
- Applying any of the trigonometric or inverse trigonometric operations from Chapters 6 and 7.

Here is the method:

- Turn the inequality 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 expression in the inequality, 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 expressions in the inequality are 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.

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