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Interval notation is a method used to describe a set of numbers along the number line. It captures the beginning and end of intervals where the inequality holds true. For inequalities, where expressions are either greater than, less than, or between values, interval notation provides a concise way to represent these conditions.

For instance, in the original exercise, we determined that the solution to \(|x-5| \geq 2\) resulted in two separate intervals: one where \((x \leq 3)\) and another where \((x \geq 7)\). In interval notation, these are written as:

• \((\-\infty, 3]\) — all numbers less than or equal to 3.
• \([7, \infty)\) — all numbers greater than or equal to 7.

The solution set of the inequality is presented as the union of these intervals, which is \((\-\infty, 3] \cup [7, \infty)\). The union symbol \((\cup)\) signifies all the numbers that belong to either of these intervals. Using interval notation helps organize and simplify expressing solution sets in mathematics.

To solve absolute value inequalities, you first need to understand what the absolute value represents. The absolute value \(|x - a|\) indicates the distance between the number \(x\) and another point \(a\) on the number line. Solving \(|x - 5| \geq 2\) means finding all values of \(x\) where the distance from 5 is at least 2.

Breaking down the inequality involves setting up two conditions because the absolute value function is always non-negative.

• For \(x - 5 \geq 2\), solve for \(x\) to get \(x \geq 7\). This reveals numbers that are 2 units or more to the right of 5.
• For \(x - 5 \leq -2\), solve for \(x\) to get \(x \leq 3\). This provides numbers that are 2 units or more to the left of 5.

The solutions to these inequalities give us the intervals that form the complete solution set. By combining them, you obtain the final solution in interval notation. The two resulting inequalities help ensure that \(x\) satisfies the original condition of being outside the range between 3 and 7.

A number line is a visual tool used to represent numbers and their relationships. When solving inequalities, drawing a number line can help you see where the solution sets lie and how they relate to each other. For absolute value inequalities like \(|x-5| \geq 2\), the number line becomes an especially valuable resource.

Start by plotting critical numbers from the solved inequalities; in this case, 3 and 7. These numbers indicate where the solution set begins or ends. Then shade the regions that satisfy each inequality:

• Shade all numbers less than or equal to 3, moving leftwards, indicating the interval \((\-\infty, 3]\).
• Shade all numbers greater than or equal to 7, moving rightwards, showing the interval \([7, \infty)\).

The number line thus provides a clear representation of how the solution comprises two separate regions, helping you understand the solution's layout intuitively. This visualization assists in verifying the intervals noted down and ensures that you have captured the essence of the inequality appropriately.