Question - Solving Absolute Value Inequality
Solution:
The inequality provided in the image is:|u + 6| ≥ 46To solve this absolute value inequality, we need to consider two separate cases because absolute value expressions represent the distance from zero and can be either positive or negative.Case 1: u + 6 is nonnegativeIf u + 6 is nonnegative, then |u + 6| just equals u + 6. So we simply remove the absolute value and solve the inequality:u + 6 ≥ 46u ≥ 46 - 6u ≥ 40Case 2: u + 6 is negativeIf u + 6 is negative, then |u + 6| equals -(u + 6). In this case, when we remove the absolute value, we need to reverse the inequality sign:-(u + 6) ≥ 46-u - 6 ≥ 46-u ≥ 46 + 6-u ≥ 52Now, we multiply both sides by -1 and remember to reverse the inequality:u ≤ -52Combining both cases, the solution to the inequality |u + 6| ≥ 46 is:u ≤ -52 or u ≥ 40This means that any value of u that is less than or equal to -52 or greater than or equal to 40 satisfies the given inequality.
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