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Absolute value is a concept that deals with the magnitude of a number, irrespective of its sign. The absolute value of a number is always non-negative. For example, the absolute value of both 3 and -3 is 3.When solving inequalities involving absolute values, such as \(|x+2| < 4\), you are looking for all numbers whose distance from -2 on the number line is less than 4. This involves considering both the positive and negative solutions:

• Start with a basic understanding: \(|a| < b\) implies that \(-b < a < b\).
• Apply this mechanism to our equation: \(|x+2| < 4\) becomes \(-4 < x+2 < 4\).

The translation into a compound inequality is a core step in solving problems involving absolute values.

A compound inequality consists of two inequalities that are joined together by a logical "and" or "or". In our case, with \(-4 < x + 2 < 4\), it's connected by "and", meaning that both inequalities must be true simultaneously.To solve a compound inequality, you typically perform the same operations on all parts. For this exercise, we subtracted 2 from each part:

• This operation keeps the inequality balanced and allows us to solve for \(x\).
• The intermediate step was: \(-4 - 2 < x < 4 - 2\).
• Simplifying results in: \(-6 < x < 2\).

This shows the range where the variable \(x\) fits, ensuring both conditions in the compound inequality are satisfied.

Interval notation is a convenient way to express a range of numbers that may satisfy an inequality. It uses parentheses or brackets to show if endpoints are included or excluded.In the context of \(-6 < x < 2\), we use round parentheses \(( )\) because -6 and 2 are not included in the solution. This tells us that the values are "approaching but not touching" -6 and 2. Therefore, the interval notation for this inequality is:

• \((-6, 2)\) signifies all numbers greater than -6 and smaller than 2.

Use square brackets \([ ]\) if the endpoints are included, like in \([a, b]\), indicating both \(a\) and \(b\) are part of the solution.Interval notation provides a clear, concise way to represent solution sets in math, making it easier to communicate the range of solutions.