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Interval notation is a way of representing subsets of real numbers on the number line. It uses parentheses, brackets, and infinity symbols to describe an interval.
- Parentheses (")" or ")") indicate that the endpoint is not included in the interval. This is called an open interval.
- Brackets ("]" or "[") suggest the endpoint is included, known as a closed interval.
When translating the solution of an inequality into interval notation, think of it as marking the start and end points on a number line. For instance, in our problem, the solution \( (-\infty, -3) \cup (7, +\infty) \) represents two separate open intervals:

• \((- \infty, -3)\) tells us that x can be any number less than -3. The negative infinity symbol \( (-\infty) \) suggests that there is no lower bound for this interval.


• \((7, +\infty)\) tells us that x can be any number greater than 7. The positive infinity symbol \( (+\infty) \) implies no upper limit.

When putting two intervals together with a union (\(\cup\)), you are showing that values can come from either interval. This notation is crucial for defining solutions for absolute value inequalities, where the solution sets may not be contiguous on a number line.

Distance on a number line is a fundamental concept that can be visualized as the absolute value between two points. When talking about distance, we’re referring to the non-negative scalar measure between numbers.
The absolute value expression \(|x - 2|\) effectively measures how far any given point x is from 2 on the number line. \(|x - 2| > 5\) implies that this distance is greater than 5 units.
Let's break this down:

• The point 2 is located somewhere on our number line.


• If we're looking for numbers more than 5 units away, we have to look at two scenarios:


• Moving to the right (x is greater than a location beyond 5 units to the right of 2, i.e., 7).


• Moving to the left (x is less than a location beyond 5 units to the left of 2, i.e., -3).

Understanding distance helps in partitioning the number line into meaningful sections which can express inequalities. Thus, noticing intervals on the number line rooted in distance measures leads us to adequate solutions of inequality problems.

Solving inequalities, especially those involving absolute values, requires thoughtful dissection into simpler parts. Absolute value inequalities like \(|x - 2| > 5\) need special handling because they involve notions of distance being more significant or smaller than a certain threshold.
The strategy often involves splitting the inequality into two simpler inequalities:

• If \(|x - 2|\) is greater than 5, it means x can either be more significant than 2 plus 5 (which is 7), giving \(x > 7\), or


• x can be less than 2 minus 5 (which is -3), giving \(x < -3\).

By solving these inequalities separately, you frame the solution around key intervals which contribute to the complete answer \( (-\infty, -3) \cup (7, +\infty) \).
Always be sure that solving these inequalities aligns with natural rules for inequalities, keeping sign changes in mind for correctly handling < or > during manipulations.These principles ensure absolute value inequalities are navigated correctly, yielding accurate interval solutions on the number line.