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In mathematics, interval notation is a way of writing subsets of the real number line. It is a concise method to represent a range of numbers by specifying the endpoints of the interval and whether or not these endpoints are included in the set.

There are two types of intervals:

• Closed interval: Both endpoints are included, denoted by square brackets \([a, b]\).
• Open interval: Neither endpoint is included, denoted by parentheses \((a, b)\).
• Half-open interval: One endpoint is included while the other is not, denoted as \([a, b)\) or \((a, b]\).

On a number line, a closed interval includes everything between and including both endpoints, while an open interval includes only the values between the endpoints. In the solved example above:

• For part (a), the solution is \([8, \infty)\), indicating 8 and all numbers greater.
• For part (b), the solution is the union of two intervals, \((\infty, 0] \cup [8, \infty)\), indicating numbers less than or equal to 0, and numbers greater than or equal to 8.
• For part (c), the interval \([0, 8]\) means all values from 0 to 8, including both.

A compound inequality is a mathematical statement involving two or more inequalities that are connected by the words "and" or "or." These types of inequalities express a range of possible solutions that satisfy one or both expressions.

For "and" compound inequalities, both conditions need to be true at the same time. They are usually written in a form like \(a < x < b\). An example from the original exercise is part (c), represented by \(-4 \leq x - 4 \leq 4\), which, when solved, translates to \(0 \leq x \leq 8\). This indicates that \(x\) must be within the intersection of the solutions, satisfying both conditions simultaneously.

In contrast, an "or" compound inequality allows for either condition to be true. This was used in step (b), where the solution to \(|x-4| \geq 4\) splits into two separate conditions: \(x \geq 8\) or \(x \leq 0\). Either of these conditions can be true, which is why the solution is expressed in interval notation as a union, \((\infty, 0] \cup [8, \infty)\). The word "or" signifies that if either inequality is true, the overall statement is true.

Absolute value inequalities involve expressions enclosed within absolute value symbols, which measure the distance a number is from zero on the number line, regardless of direction.

In absolute value inequalities, you'll typically see two forms:

• Inequality of the form \( |expression| < k \): This means that the expression is less than \(k\) units away from zero, providing a range of solutions between two endpoints. For example, the inequality \(|x - 4| \leq 4\) results in the compound inequality \(-4 \leq x-4 \leq 4\), leading to \(0 \leq x \leq 8\). This suggests the values of \(x\) that reside within 4 units of 4.
• Inequality of the form \( |expression| > k \): This shows that the expression is more than \(k\) units from zero, resulting in two separate solutions. In the solved example, \(|x - 4| \geq 4\) indicates that the distance between \(x\) and 4 is greater than or equal to 4, hence resulting in solutions \(x \geq 8\) or \(x \leq 0\).

Understanding these concepts helps solve absolute value inequalities by setting up two potential inequalities—one for the positive scenario and one for the negative scenario, ensuring all possible solutions are covered.