91Ó°ÊÓ

Interval notation is a concise way to represent a range of values, particularly useful when dealing with inequalities. Instead of using words, interval notation employs brackets and parentheses to describe the set of solutions.

There are a few different symbols to know:

• Parentheses \( () \) express that an endpoint is not included in the set.
• Brackets \[ [] \] signify that an endpoint is part of the set.

For example, the interval notation \( (2, 5] \) includes numbers greater than 2 but less than or equal to 5. Notice how 5 has a bracket because it is included in the set, while 2 has a parenthesis because it is excluded.

In equality problems like our original exercise \((x-2)^2 > 0\), the solution is indicated as \((- ext{∞}, 2) \cup (2, ext{∞})\). This tells us that the solution includes all real numbers except 2, which is represented by using parentheses around the 2.

In mathematics, an expression is a combination of numbers, variables, and operations (like addition or multiplication) that represents a value. Understanding expressions is crucial as they form the backbone of equations and inequalities.

Expressions can be as simple as a single number or variable, such as 5 or \(x\), or as complex as \((x-2)^2\). The expression \((x-2)^2\) plays a central role in our problem. It means that the value of \(x-2\) is squared, or multiplied by itself. A key property of squares of real numbers is that they are always non-negative. This is because multiplying two negative numbers or two positive numbers always results in a positive number.

Another important concept is evaluating expressions. This means calculating the number the expression represents for specific values of variables. In our problem, evaluating \(x=2\) in the expression \((x-2)^2\) results in zero, while any other real number results in a positive number.

Real numbers encompass all the numbers you can think of on the number line. This includes:

• Natural numbers (1, 2, 3, ...).
• Whole numbers (0, 1, 2, 3, ...).
• Integers (..., -2, -1, 0, 1, 2, ...).
• Rational numbers (fractions like 1/2, 3/4, ...).
• Irrational numbers (numbers that cannot be expressed as fractions, such as \( \sqrt{2} \) or \( \pi \)).

In our inequality problem, the term "real numbers" is mentioned when expressing the solution where \(x \) can take on any value except 2. This entails that any value on the number line, apart from 2, will make \((x-2)^2 > 0\) true.

Knowing that any real number can be included or excluded as needed allows us to precisely define solution sets to these kinds of mathematical problems. The real numbers provide a complete picture of possible values, from negative to positive infinity, offering a comprehensive understanding of where our expressions hold valid solutions.