To solve the inequality [tex]\( |4w - 2| - 6 > 8 \)[/tex], we'll follow these steps:
### Step 1: Isolate the absolute value expression
First, we'll isolate the absolute value term by adding 6 to both sides of the inequality:
[tex]\[ |4w - 2| - 6 > 8 \][/tex]
[tex]\[ |4w - 2| > 14 \][/tex]
### Step 2: Solve the absolute value inequality
The absolute value inequality [tex]\( |4w - 2| > 14 \)[/tex] can be split into two separate inequalities:
1. [tex]\( 4w - 2 > 14 \)[/tex]
2. [tex]\( 4w - 2 < -14 \)[/tex]
### Step 3: Solve each inequality separately
Next, we'll solve both inequalities:
#### Inequality 1: [tex]\( 4w - 2 > 14 \)[/tex]
[tex]\[ 4w - 2 > 14 \][/tex]
Add 2 to both sides:
[tex]\[ 4w > 16 \][/tex]
Divide both sides by 4:
[tex]\[ w > 4 \][/tex]
#### Inequality 2: [tex]\( 4w - 2 < -14 \)[/tex]
[tex]\[ 4w - 2 < -14 \][/tex]
Add 2 to both sides:
[tex]\[ 4w < -12 \][/tex]
Divide both sides by 4:
[tex]\[ w < -3 \][/tex]
### Step 4: Combine the solutions
The solutions to the inequalities are [tex]\( w > 4 \)[/tex] or [tex]\( w < -3 \)[/tex].
### Conclusion
Thus, the solution to the inequality [tex]\( |4w - 2| - 6 > 8 \)[/tex] is:
[tex]\[ w > 4 \quad \text{or} \quad w < -3 \][/tex]
This means that [tex]\( w \)[/tex] falls into the intervals [tex]\( (-\infty, -3) \)[/tex] or [tex]\( (4, \infty) \)[/tex].
So the correct answer is:
[tex]\[ w > 4 \) or \( w < -3 \][/tex]
