Question - Solving an Inequality Involving Absolute Value
mathabsolute value expressioninequality solution stepsinequality involving absolute valuesolve absolute value inequality
Solution:
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 $$ s > \frac{1}{2} $$.
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