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Interval notation is a way to describe a range of numbers, typically the solutions to inequalities. When we solve inequalities, we often find that there are many possible solutions that form a continuous range. Interval notation describes this range compactly and clearly.
For example, if we have a solution that says all numbers between 1 and 5, we write it as \((1, 5)\). Note that we use parentheses because these symbols indicate that the endpoints 1 and 5 are not included in the interval. If they were included, we would use square brackets: \([1, 5]\).

• Round brackets \((a, b)\) indicate endpoints are not included.
• Square brackets \([a, b]\) indicate endpoints are included.
• Mixed brackets \([a, b)\) or \((a, b]\) mean one endpoint is included and one is not.

Using interval notation is important because it provides a clear and concise way to express a set of solutions for inequalities. It's a universal language in mathematics to articulate ranges, which can also be used for functions, domain, and range analysis.

Reciprocal functions are those of the form \(f(x) = \frac{1}{x}\). These functions have unique behaviors that are essential to grasp when solving inequalities involving them.
One key feature of a reciprocal function is that as \(x\) approaches zero, \(f(x)\) will become very large or very small depending on the direction of approach.

• If \(x\) approaches zero from the positive side, \(f(x)\) approaches positive infinity.
• If \(x\) approaches zero from the negative side, \(f(x)\) approaches negative infinity.

This sensitivity near zero means that reciprocal functions do not exist at \(x = 0\) and are undefined there. Also, reciprocating a number flips the sign of the inequality when solving. For example, if we start with \(\frac{1}{x} > 2\), flipping \(1/x\) to solve for \(x\) gives us \(x < \frac{1}{2}\). Understanding this flip is crucial when handling reciprocals in inequality problems.

Solving inequalities involves a methodical approach, often similar to solving equations but with added steps to account for the direction of inequalities. Here's a simple approach using the given types of inequalities:

• Isolate the reciprocal: For inequalities like \(1.99 < \frac{1}{x} < 2.01\), start by considering the reciprocal expressions and separate them into two simpler inequalities.
• Flip and solve: Reciprocals are unique as solving them requires flipping the inequality sign when switching sides. Consider \(\frac{1}{x} > 1.99\), reciprocal it to find \(x < \frac{1}{1.99}\).
• Merge solutions: Once solved, compile these individual solutions into an interval. For example, solutions \(x > \frac{1}{2.01}\) and \(x < \frac{1}{1.99}\) combine to form the interval \((\frac{1}{2.01}, \frac{1}{1.99})\).
• Express in interval notation: Finally, express the solution as a clear interval, using the correct brackets to indicate included or excluded endpoints.

Each step ensures you respect the mathematical properties of inequalities and reciprocal functions, leading to the correct solution in an easy-to-understand interval notation.