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Interval notation is a succinct way to display the range of values that satisfy an inequality. It uses brackets and parentheses to describe intervals:

• Square brackets "\([\ ]\)" denote that the endpoint is included in the set, meaning the solution is "greater than or equal to" (\(\geq\)) or "less than or equal to" (\(\leq\)).
• Parentheses "\((\ )\)" indicate that the endpoint is not included, meaning the solution only satisfies "greater than" (\(>\)) or "less than" (\(<\)).

To better understand this, let's look at our original inequality expressed as \(-\frac{1}{2} \leq x < \frac{2}{3}\). In interval notation, this translates into:\[[-\frac{1}{2}, \frac{2}{3} )\]This symbolically communicates that the values of \(x\) include \(-\frac{1}{2}\) (since it's a closed interval at this endpoint due to the inequality being \(\leq\)) and go up to but do not include \(\frac{2}{3}\) (as indicated by the \(<\) result in an open interval). Understanding interval notation simplifies comparing and communicating solutions of inequalities.

A compound inequality consists of two separate inequalities combined into one statement by the word "and" or "or".

• "And" indicates that both conditions must be true simultaneously. For example, \(-3 < 1 - 6x \leq 4\) is a compound inequality because it requires \(x\) to fulfill both parts of the statement.
• Notice how this particular inequality can be broken down into two parts, \(-3 < 1 - 6x\) and \(1 - 6x \leq 4\), and this means we are looking at the overlap or intersection where both are satisfied.

In solving the compound inequality for \(x\), you must handle both restrictions together, performing operations across all parts. Once simplified, the solution \(-\frac{1}{2} \leq x < \frac{2}{3}\) results, combining these conditions and requiring \(x\) to lie in this specific interval. This preserve the logical requirement of simultaneously meeting both inequalities.

Graphing inequalities on a number line visually showcases the solution set and its boundaries. This method involves a few easy steps:

• Firstly, identify whether endpoints are included. For \(-\frac{1}{2}\) which is included, use a solid dot. Conversely, \(\frac{2}{3}\) isn't included, so you use an open circle.
• Then, shade the region between \(-\frac{1}{2}\) and \(\frac{2}{3}\), indicating all the values \(x\) can take.

Sketching it on a number line helps solidify where the solutions of the inequality lie. This visualization complements interval notation by making it easy to see which parts of the number line are part of the solution, thus bridging the gap between abstract symbols and concrete understanding.