Answer
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Step 1:I'll solve this absolute value inequality step by step:
Step 2:: Understand the Inequality
The inequality $$\left|x-6\right| \geq 5$$ means the distance between x and 6 on a number line is greater than or equal to 5.
Step 3:: Solve the Inequality
Case 2: $$x - 6 \leq -5
This means x can be 5 or more units to the left of 6, or 5 or more units to the right of 6. Mathematically, this splits into two cases:
Step 4:: Solve Case 1
x - 6 \geq 5
x \geq 11
Step 5:: Solve Case 2
x - 6 \leq -5
x \leq 1
Step 6:: Combine the Solutions
The solution is the union of these two intervals: $$x \in (-\infty, 1] \cup [11, \infty)
Final Answer
The solution set is all real numbers less than or equal to 1, or greater than or equal to 11, which can be written as x \in (-\infty, 1] \cup [11, \infty).
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