Solving Absolute Value Inequality
The inequality provided in the image is:
|u + 6| ≥ 46
To 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 nonnegative
If u + 6 is nonnegative, then |u + 6| just equals u + 6. So we simply remove the absolute value and solve the inequality:
u + 6 ≥ 46
u ≥ 46 - 6
u ≥ 40
Case 2: u + 6 is negative
If 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 ≥ 52
Now, we multiply both sides by -1 and remember to reverse the inequality:
u ≤ -52
Combining both cases, the solution to the inequality |u + 6| ≥ 46 is:
u ≤ -52 or u ≥ 40
This 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.
