Problem 85 - College Algebra Essentials

Question

Solve each absolute value inequality. $$3 \leq|2 x-1|$$

Step-by-Step Solution

Answer

The solution of the inequality \(3 \leq |2x - 1|\) is \(x \geq 2\) or \(x \leq -1\)

Step 1: Understand the Absolute Value Inequality

The absolute value equation can represent two different linear equations. The inequality \(3 \leq |2x - 1|\) can give two scenarios - one where \(2x - 1 \geq 3\) and the other where \(2x - 1 \leq -3\). Both cases must be evaluated individually.

Step 2: Solve the First Linear Inequality

The first inequality is \(2x - 1 \geq 3\). To solve this for x, start by adding 1 on both sides to eliminate the constant from the left-hand side. This results in \(2x \geq 4\). Then, to isolate x, divide both sides by 2. This gives \(x \geq 2\). So for the first case, any value of x which is equal to or greater than 2 is a solution.

Step 3: Solve the Second Linear Inequality

The second inequality is \(2x - 1 \leq -3\). To solve this inequality, add 1 to both sides, resulting in \(2x \leq -2\). After that, divide both sides by 2 to isolate x. This gives \(x \leq -1\). For the second case, any value of x which is equal to or less than -1 is a solution.

Step 4: Combine the Solutions

The final solution to the problem is found by combining the solutions for the two cases. The solution for the absolute value inequality \(3 \leq |2x - 1|\) is all \(x\) such that \(x \geq 2\) or \(x \leq -1\)