91Ó°ÊÓ

A number line is a visual representation of numbers along a straight horizontal line. It's a great tool to understand and solve inequalities. In a number line, each point corresponds to a number. For the compound inequality in this exercise, you will plot individual inequalities first.

To graph \(x > -2\), place an open circle on \(-2\) and shade all the points to the right because all numbers greater than \(-2\) are valid. To graph \(x < 4\), place an open circle on \(4\) and shade all points to the left because all numbers less than \(4\) are valid. Combining both, the overlapping region between \(-2\) and \(4\) is shaded.

Remember, open circles indicate that the endpoint is not included in the solution set.

Interval notation is a way to describe the solution set of an inequality using intervals. It's concise and straightforward. For example, the interval \((-2, 4)\) means every number between \(-2\) and \(4\), excluding \(-2\) and \(4\) themselves.

Different brackets and parentheses are used in interval notation:

• \textbf{Parentheses}: \( ( \, \ ) \) indicate that the endpoints are not included.
• \textbf{Brackets}: \( [ \, \ ] \) indicate that the endpoints are included.

For the compound inequality \(x > -2\) and \(x < 4\), we use parentheses: \((-2, 4)\). This tells us that the numbers \(-2\) and \(4\) are not part of the solution set.

Inequalities are mathematical expressions involving the symbols >, <, \(\textbackslash ge\), or \(\textbackslash le\). They show the relationship between two values, indicating that one is larger or smaller than the other.

In this exercise, we deal with compound inequalities, which combine two inequalities using the words ‘and’ or ‘or’. Here both conditions must be true simultaneously, expressed with ‘and’.

Simply put, a compound inequality like \(x > -2\) \(\text{and}\) \(x < 4\) means that \(x\) must be greater than \(-2\) and less than \(4\) at the same time. Graphing this on a number line and writing it in interval notation helps visualize and understand the solution set effectively.