Solve the absolute value inequality.
2\
|x+4|- 1≥9
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
• A. The solution set is
(Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)
• B. The solution is the empty set.
To solve the inequality 2|x+4| - 1 ≥ 9, we can isolate the absolute value term and then solve for x.
Adding 1 to both sides of the inequality, we get:
2|x+4| ≥ 10
Next, dividing both sides of the inequality by 2, we have:
|x+4| ≥ 5
Since the inequality involves absolute value, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: x + 4 ≥ 0
If x + 4 ≥ 0, then |x+4| simplifies to x + 4. So we have:
x + 4 ≥ 5
Solving for x, we subtract 4 from both sides:
x ≥ 1
Case 2: x + 4 < 0
If x + 4 < 0, then |x+4| simplifies to -(x + 4). So we have:
-(x + 4) ≥ 5
To remove the negative sign, we multiply both sides of the inequality by -1:
x + 4 ≤ -5
Solving for x, we subtract 4 from both sides:
x ≤ -9
Putting the results of both cases together, we have the solution set:
x ≤ -9 or x ≥ 1
In interval notation, this can be written as:
(-∞, -9] ∪ [1, ∞)
Therefore, the correct choice is:
A. The solution set is (-∞, -9] ∪ [1, ∞)