Problem 73 - College Algebra Essentials

Question

Solve each absolute value inequality. $$|3 x-8|>7$$

Step-by-Step Solution

Answer

The solution is \(x < 1/3\) or \(x > 5\).

Step 1: Set up two separate inequalities

The first step is to set up two inequalities we can solve independently. Because \(|3x - 8| > 7\), it means 3x - 8 is either greater than 7 or less than -7. So we set up these two inequalities: \(3x - 8 > 7\) and \(3x - 8 < -7\).

Step 2: Solve the first inequality

Solve the first inequality \(3x -8 > 7\). Start by adding 8 to both sides to get \(3x > 15\). Then divide each side by 3 to get \(x > 5\).

Step 3: Solve the second inequality

Next, solve the second inequality \(3x - 8 < -7\). Start by adding 8 to both sides to get \(3x < 1\). Then divide each side by 3 to isolate the x, which gives \(x < 1/3\).

Step 4: Write the final solution

Finally, we combine the solutions of the two separate inequalities to formulate the complete solution. The solutions are \(x > 5\) and \(x < 1/3\), meaning the values of x that satisfy \(|3x - 8| > 7\) are either less than 1/3 or greater than 5.