Problem 79 - College Algebra Essentials

Question

Solve each absolute value inequality. $$3|x-1|+2 \geq 8$$

Step-by-Step Solution

Answer

The solution to the inequality \(3|x-1|+2 \geq 8\) is \(x \in (-\infty,-1] \cup [3,\infty)\)

Step 1: Isolate the Absolute Value

First, isolate the absolute value on one side of the inequality by subtracting 2. The inequality becomes \(3|x-1| \geq 6\). Then divide each side by 3 to get \(|x-1| \geq 2\).

Step 2: Split the Inequality into Two Cases

An absolute value inequality can be split into two separate inequalities. We have \(x-1 \geq 2\) and \(x-1 \leq -2\).

Step 3: Solve Each Case

Solve each inequality on its own. For \(x-1 \geq 2\), the solution is \(x \geq 3\). For \(x-1 \leq -2\), the solution is \(x \leq -1\).

Step 4: Express the Solution

The two inequalities represent different sections of the number line. Therefore, the solution is the union of two intervals, which we denote by \(x \in (-\infty,-1] \cup [3,\infty)\)