91Ó°ÊÓ

Compound inequalities involve two separate inequalities that are connected by either "and" or "or." In the expression \(-4 < x < 4\), the statement tells us that \(x\) must satisfy both conditions simultaneously: it must be greater than \(-4\) and less than \(4\). This is known as a conjunction, which uses the "and" operator.Conjunctions indicate that the solution must satisfy all conditions at once. When dealing with compound inequalities, it’s often helpful to break them down into their individual parts before finding the overlap. For example, the inequality \(-4 < x < 4\) can be seen as the overlap of \(-4 < x\) and \(- x < 4\). This overlap defines the range of values that \(x\) can take. Compound inequalities like this are useful for defining ranges on a number line, which will come in handy when plotting and solving real problems.

Interval notation is a concise way of expressing a set of numbers along a number line. It shows the set of solutions as an interval, using parentheses \(()\) and brackets \([]\). Parentheses mean that the endpoints are not included, while brackets mean they are included.For the inequality \(-4 < x < 4\), the solution set is \((-4, 4)\). Since the inequality does not include the endpoints \(-4\) and \(4\) (as indicated by the strict inequality '<'), we use parentheses. Thus, \((-4, 4)\) represents all numbers greater than \(-4\) and less than \(4\), but not actually \(-4\) or \(4\) themselves.Interval notation makes it easy to read and write inequalities quickly, and is particularly useful when dealing with continuous ranges of solutions.

• A closed interval like \([a, b]\) means \(x\) includes both \(a\) and \(b\).
• An open interval like \((a, b)\) features neither \(a\) nor \(b\).
• Mixed intervals, such as \((a, b]\), include only one endpoint.