Solving an Inequality Involving Absolute Value
To solve the inequality \( 8 - 7|1 - 6s| < -6 \), we could follow these steps:
1. First, let's simplify the inequality by removing the constant term '8' from both sides:
\[ 8 - 7|1 - 6s| - 8 < -6 - 8 \]
\[ -7|1 - 6s| < -14 \]
2. Now, we divide both sides by '-7'. Remember that dividing by a negative flips the inequality sign:
\[ \frac{-7|1 - 6s|}{-7} > \frac{-14}{-7} \]
\[ |1 - 6s| > 2 \]
3. The inequality \( |1 - 6s| > 2 \) means that the expression inside the absolute value is either greater than 2 or less than -2. We can express this as two separate inequalities:
\[ 1 - 6s > 2 \quad \text{or} \quad 1 - 6s < -2 \]
4. To solve \( 1 - 6s > 2 \), subtract '1' from both sides and then divide by '-6':
\[ -6s > 1 \]
\[ s < -\frac{1}{6} \]
5. To solve \( 1 - 6s < -2 \), subtract '1' from both sides and then divide by '-6':
\[ -6s < -3 \]
\[ s > \frac{1}{2} \]
The solution to the inequality is therefore \( s < -\frac{1}{6} \) or \( s > \frac{1}{2} \).
