Problem 83 - College Algebra Essentials

Question

Solve each absolute value inequality. $$-4|1-x|<-16$$

Step-by-Step Solution

Answer

\(x \in (-\infty,-3) \cup (5,\infty)\)

Step 1: Isolate the Absolute Value

One could start with isolating the absolute value by dividing both sides of the inequality by -4, remembering to reverse the direction of the inequality because negative number division was involved and that changes inequality direction. Hence, the inequality becomes \(|1-x| > 4\).

Step 2: Split the Absolute Value

After isolating the absolute value, split it up into two separate inequalities, one for the positive case and one for the negative. The resulting system of inequalities will be \(1-x > 4\) and \(1-x < -4\).

Step 3: Solve the Two Inequalities

Next, solve each inequality separately. For \(1-x > 4\), subtract 1 from both sides to get \(-x > 3\), then multiply by -1 (and flip the inequality) to get \(x < -3\). For \(1-x < -4\), subtract 1 from both sides to get \(-x < -5\), then multiply by -1 (and flip the inequality) to get \(x > 5\).

Step 4: Express the Solution

The solution is the set of all x that satisfy either inequality. In interval notation, that would be \(x \in (-\infty,-3) \cup (5,\infty)\) as this is the union of two disjoint intervals.