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Interval notation is a concise way of writing sets of numbers, often used to describe the solution sets of inequalities. It uses parentheses and brackets to show which numbers are included or excluded in a set. This notation is especially helpful when dealing with ranges of values.

For example, consider the interval \((-3, 5]\). The round parenthesis \(()\) at \(-3\) indicates that \(-3\) is not included in the interval. The square bracket \(()\) at \(5\) indicates that \(5\) is included.

• An open interval like \((-3, 5)\) means neither boundary is included.
• A closed interval like \([2, 6]\) means both endpoints are included.
• A half-open interval like \((3, 7]\) means \(3\) is not included, but \(7\) is.

In our example, the solution for the inequality is \((-\frac{26}{3}, \frac{16}{3}]\). This means all numbers \(x\) from slightly greater than \(-\frac{26}{3}\) to \(\frac{16}{3}\), including \(\frac{16}{3}\), satisfy the inequality.

Compound inequalities involve two or more inequalities combined by the words 'and' or 'or'. In the exercise, we see a double inequality: \[4 > \frac{2-3x}{7} \geq -2\] which is a typical example of a compound inequality. These inequalities demand a solution that satisfies all parts of the system.

• 'And' compound inequalities: Require both conditions to be true simultaneously. This often results in an intersection of solutions.
• 'Or' compound inequalities: Require at least one condition to be true. This typically leads to a union of solutions.

In our problem, we dealt with an 'and' situation. We first broke down the compound inequality into simpler parts that were easier to handle individually. After solving each part, we combined them, ensuring the solution satisfied both conditions. Therefore, the final solution reflects numbers that fit into both sets of conditions.

When faced with solving inequalities, a step-by-step approach breaks the process into manageable parts. This guided approach assists in keeping track of the problem and detecting errors before they compound.

Here’s a brief guide to the method:

• **Break the problem into parts**: As in our exercise, separate the compound inequality into two parts. This simplifies the problem.
• **Solve each part**: Figure out each inequality separately. This involves operations like multiplying, dividing, or rearranging terms. Be vigilant with inequalities: Remember to flip the sign when multiplying or dividing by a negative number.
• **Combine solutions**: Bring all parts back together and find the overlap, often visualized on a number line. Since we had an 'and' inequality, our solution is the intersection.
• **Express in interval notation**: Once satisfied with the solution, express it in the neat form of interval notation, like \((-\frac{26}{3}, \frac{16}{3}]\).

These steps, when followed meticulously, can simplify solving even complex inequality problems and sharpen algebraic skills over time.