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Interval notation is a concise way to represent a range of numbers on the number line. This system is particularly useful when dealing with solutions to inequalities, as it captures all the numbers within a certain interval that satisfy the inequality.

In interval notation, we use brackets and parentheses to denote closed and open intervals, respectively. A closed interval, indicated by square brackets \[a, b\], means that the endpoints a and b are included in the set. An open interval, indicated by parentheses \((a, b)\), means the endpoints are not included.

For solutions extending indefinitely in one direction, the symbols \(-\infty\) (negative infinity) and \(+\infty\) (positive infinity) are used. These are always accompanied by parentheses, as infinity itself can never be reached or included. Combining intervals with a union (\(\cup\)) allows us to represent non-adjacent ranges. This is exactly what we see in the provided exercise solution, where the solution set includes all numbers less than or equal to -8 and all numbers greater than or equal to 2.

Understanding how to find solutions to inequalities is crucial in mathematics, particularly when dealing with absolute value equations. An inequality, unlike an equation, can have a wide range of solutions, often expressed as intervals.

When solving an inequality, we perform similar operations as with an equation, except we must be careful when multiplying or dividing by a negative number, which reverses the inequality sign. The solution to an inequality represents all the values that satisfy the inequality, not just a single number.

In the context of the exercise, we encounter an absolute value inequality which gives rise to two separate inequalities. Each is solved independently to determine the solution set. It's essential to remember that these solutions signify where the original inequality holds true and they are translated into interval notation to clearly communicate the complete range of solutions.

An absolute value equation involves the absolute value function, which measures the distance of a number from zero on the number line. The absolute value of a number is non-negative, regardless of whether the input is negative or positive.

The key to solving absolute value equations is to recognize that if \(|x| = a\), where \(a\) is a positive number, then \(x\) can either be \(a\) or \(-a\). This leads to two separate scenarios to consider. Similarly, for an absolute value inequality such as \(|x+3| \geq 5\), we consider the cases where \(x+3\) is either greater than or equal to 5 or less than or equal to -5.

This approach splits the problem into two more manageable parts and allows us to solve for the variable as shown in the step by step solution. Once solved, the individual inequalities provide intervals that make up the final solution set for the original inequality.