Final answer:
To solve the inequality |x-2|+5>=12 in interval notation, isolate the absolute value expression and consider two cases when the expression inside the absolute value is positive and negative. Then, combine the solutions from both cases to get the interval notation solution.
Step-by-step explanation:
To solve the inequality |x-2|+5>=12 in interval notation, we need to isolate the absolute value expression. First, subtract 5 from both sides to get |x-2| >= 7. Next, consider the two possible cases when the expression inside the absolute value is positive and negative. When x-2>=0 (or x>=2), the absolute value expression simplifies to x-2 >= 7. Solve this inequality to get x >= 9. When x-2<0 (or x<2), the absolute value expression becomes -(x-2) >= 7. Solve this inequality by multiplying both sides by -1 (which flips the inequality sign) to get x-2 <= -7. Add 2 to both sides to get x <= -5.
Combining the solutions from both cases, we have two intervals: [9, ∞) and (-∞, -5]. Therefore, the solution in interval notation is (-∞, -5] U [9, ∞).