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Compound inequalities involve two separate inequalities that are combined into one statement by the words 'and' or 'or'. They often appear when dealing with absolute value inequalities. For example, the absolute value inequality \( |p-2| < 3 \) can be broken down into a compound inequality \( -3 < p - 2 < 3 \).

We handle compound inequalities by solving each part of the inequality separately and then combining the results. In our case, this means solving for \( -3 < p - 2 \) and \( p - 2 < 3 \).

After solving each part, add the results together to find the complete solution: \( -1 < p < 5 \). This integrated approach ensures that the solution satisfies both parts of the original inequality.

Set-builder notation is a precise way of describing a set by stating a property that its members must satisfy. For example, the solution \( -1 < p < 5 \) can be written in set-builder notation as \( \{ p \mid -1 < p < 5 \} \).

This notation includes a variable, a vertical bar (or 'such that'), and a condition. In set-builder notation, the set comprises all elements that satisfy the given condition. For our inequality, it means finding all values of \( p \) that lie between -1 and 5.

This concise mathematical language is helpful when solving or representing complex sets of solutions.

Interval notation is another compact method to express the range of solutions for inequalities. Instead of words or phrases, it uses brackets and parentheses to describe intervals.

For instance, the solution \( -1 < p < 5 \) translates to \( (-1, 5) \) in interval notation. The parentheses indicate that the endpoints are not included within the interval.

Different symbols mean different things:

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