Solve the absolute value inequality. Express the answer using interval notation.
9 − |2x + 7| ≤ 6
First, let's isolate the absolute value expression by subtracting 9 from both sides:
-|2x + 7| ≤ 6 - 9
-|2x + 7| ≤ -3
Next, let's remove the negative sign by multiplying through by -1:
|2x + 7| ≥ 3
Now, we can split the absolute value into two separate cases:
Case 1: (2x + 7) ≥ 3
2x + 7 ≥ 3
2x ≥ -4
x ≥ -2
Case 2: -(2x + 7) ≥ 3
-2x - 7 ≥ 3
-2x ≥ 10
x ≤ -5
Therefore, the solution to the inequality is x ≤ -5 or x ≥ -2, which can be expressed using interval notation as (-∞, -5] ∪ [-2, ∞).
To solve the absolute value inequality 9 - |2x + 7| ≤ 6, we will consider two cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: 2x + 7 ≥ 0
In this case, the absolute value simplifies to:
9 - (2x + 7) ≤ 6
2x + 7 ≥ 3
Subtracting 7 from both sides:
2x ≥ -4
Dividing both sides by 2 (since 2 is positive):
x ≥ -2
Case 2: 2x + 7 < 0
In this case, the absolute value simplifies to:
9 - (-(2x + 7)) ≤ 6
9 + 2x + 7 ≤ 6
2x + 16 ≤ 6
Subtracting 16 from both sides:
2x ≤ -10
Dividing both sides by 2:
x ≤ -5
Now, we combine the solutions from both cases:
x ≥ -2 or x ≤ -5
Therefore, the solution expressed in interval notation is:
(-∞, -5] U [-2, ∞)