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Solving inequalities is a fundamental part of algebra that requires understanding how to manipulate mathematical expressions to find the set of all possible solutions. In the case of absolute value inequalities like \(|x-5| \geq 12\), we start by recognizing that the expression can have two distinct scenarios due to the nature of absolute values. This inequality indicates that the distance between \(x\) and 5 must be at least 12 units.

To solve \(|x-5| \geq 12\), split the inequality into two separate inequalities: \(x - 5 \geq 12\) and \(x - 5 \leq -12\). Solving these inequalities individually gives two sets of solutions. For the first part, add 5 to both sides to get \(x \geq 17\). For the second, also add 5 to both sides, yielding \(x \leq -7\). Thus, the solutions to \(|x-5| \geq 12\) are the values of \(x\) that satisfy either \(x \geq 17\) or \(x \leq -7\).

• Start by understanding the problem: Look for what the absolute value expression implies.
• Split the inequality based on the absolute value definition.
• Solve each inequality separately for a clearer picture of the solution set.

Once we've solved the absolute value inequalities and obtained solutions for \(x\) that are either \(x \geq 17\) or \(x \leq -7\), it's time to express these results using interval notation. Interval notation is a concise way of writing a range of numbers, which is especially handy when dealing with sets of solutions from inequalities.

In our situation, we have two separate regions that satisfy the inequality:

• The numbers less than or equal to -7: This is written as \((-\infty, -7]\).
• The numbers greater than or equal to 17: This solution is represented as \[17, \infty)\].

Interval notation uses brackets to indicate whether endpoints are included or excluded:- Curly brackets \([ ]\) include the number itself.- Round brackets \( )(\) exclude the number.By combining these two intervals with a union symbol (\

Step-by-step solutions are a vital tool in learning algebra, as they break down complex problems into more manageable parts. When tackling absolute value inequalities, each step focuses on understanding a particular aspect of the inequality.

For example, solving \( |x-5| \geq 12 \):1. **Understand the absolute value inequality**: Recognize what the inequality means in terms of distance and direction.2. **Split the inequality**: Break down the absolute value inequality into two linear inequalities using the rule \( |A| \geq B \) implies \( A \geq B \) or \( A \leq -B \).3. **Solve each inequality**: Manipulate each inequality to isolate \(x\) and solve.4. **Integrate solutions**: Merge the two solution sets descriptively using interval notation.

These steps not only help in finding the solution but also build a deeper understanding of algebraic concepts. As you consistently practice step-by-step methods, you'll become more confident in solving not only inequalities but also other algebraic problems. These meticulous approaches eventually lead to stronger problem-solving skills and analytical thinking.