Question - Solving Absolute Value Inequalities
mathabsolute value equationssolving absolute value inequalitiesabsolute value inequalitiesinequality solutions
Solution:
To solve the inequality $$ 8 - 7|1 - 6s| < -6 $$, follow these steps:1. First, isolate the absolute value term on one side of the inequality:$$ 8 - 7|1 - 6s| + 6 < 0 $$$$ 14 - 7|1 - 6s| < 0 $$$$ -7|1 - 6s| < -14 $$Now divide by -7, remembering that dividing by a negative number will flip the inequality sign:$$ |1 - 6s| > 2 $$2. Next, we need to consider the two cases due to the absolute value, where $$1 - 6s$$ can be either greater than 2 or less than -2.**Case 1 - Positive case:**$$ 1 - 6s > 2 $$Subtract 1 from both sides:$$ -6s > 1 $$Divide by -6 and flip the inequality sign:$$ s < \frac{1}{6} $$**Case 2 - Negative case:**$$ 1 - 6s < -2 $$Subtract 1 from both sides:$$ -6s < -3 $$Divide by -6 and flip the inequality sign:$$ s > \frac{1}{2} $$3. Combine both cases to get the final solution:$$ s < \frac{1}{6} $$ or $$ s > \frac{1}{2} $$The solution to the inequality is all $$ s $$ that satisfy either $$ s < \frac{1}{6} $$ or $$ s > \frac{1}{2} $$.
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