Question - Solving an Inequality Involving Absolute Value
Solution:
To solve the given inequality, $$ 8 - 7|-6s| < -6 $$, let's first simplify the expression inside the absolute value sign by multiplying 7 and the absolute value of $$-6s$$:\[ 8 - 7 \cdot |-6s| < -6 \]Let $$ A = |-6s| $$, then we have:\[ 8 - 7A < -6 \]Now, let's solve for $$ A $$ by isolating it on one side:\[ -7A < -6 - 8 \]\[ -7A < -14 \]Divide both sides by -7, and remember to reverse the inequality sign since we are dividing by a negative number:\[ A > 2 \]But $$ A $$ was defined as $$ |-6s| $$, so we substitute back in:\[ |-6s| > 2 \]Now we have to consider the two cases due to the absolute value. This inequality splits into two separate inequalities:1. When the expression inside the absolute value is positive or zero:\[ -6s > 2 \]\[ s < -\frac{1}{3} \]2. When the expression inside the absolute value is negative:\[ -6s < -2 \]\[ s > \frac{1}{3} \]Since no values of $$ s $$ can be simultaneously less than $$-\frac{1}{3}$$ and greater than $$\frac{1}{3}$$, this inequality has no solution. It means there are no values of $$ s $$ that can satisfy the original inequality $$ 8 - 7|-6s| < -6 $$.
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