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Inequalities are similar to equations, but instead of finding numbers that make two expressions equal, you find numbers that satisfy a particular relationship, like less than or greater than. Solving inequalities involves finding all values for a variable that make the inequality true. The process is much like solving equations; however, there's a special rule for multiplication or division by a negative number, which reverses the inequality sign. For the inequality given, \(-5 < t - 2\), our goal was to isolate the variable \(t\). Starting with this expression, we added 2 to both sides, maintaining the overall balance. Adding or subtracting numbers in inequalities doesn't change the inequality sign. That brought us to \(-3 < t\), indicating any number greater than \(-3\) satisfies the inequality. Checking with test values confirms if our steps are correct.

Prealgebra is much like the foundation of a sturdy building; it sets the basis for all future math learning. At this stage, you learn the basics of numbers, including operations like addition, subtraction, multiplication, and division. Understanding relationships between numbers through concepts such as inequalities is crucial. In our inequality \(-5 < t - 2\), prealgebra teaches you to solve it using fundamental operations like addition or subtraction to isolate variables. You learn the importance of performing the same operation on both sides to maintain balance. This stage also involves the logical inference where choosing test values such as \(t=0\) can verify if the solution is correct, reinforcing the prealgebraic concept of validation.

Interval notation is a concise way of describing a set of numbers. This is often used when solving inequalities because it provides a clear picture of all possible solutions at a glance. When the inequality \(-3 < t\) has been solved, it means \(t\) is any number greater than \(-3\). In interval notation, this is expressed as \((-3, \infty)\), which indicates all real numbers greater than \(-3\) up to positive infinity, without including \(-3\) itself. This notation is powerful and widely used in calculus and algebra for specifying domains, ranges, or solutions succinctly. Remember, parentheses \(()\) indicate that the endpoints are not included, while brackets \([]\) would signify included endpoints.