Problem 91 - College Algebra Essentials

Question

Solve each absolute value inequality. $$12<\left|-2 x+\frac{6}{7}\right|+\frac{3}{7}$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < -5.5\), \(x > 6\).

Step 1: Simplify the absolute value inequality

First, let's isolate the absolute value on one side of the inequality. We do it by subtracting \(\frac{3}{7}\) from both sides. So the inequality becomes: \(12 - \frac{3}{7}<|-2x+\frac{6}{7}|\).

Step 2: Solve the equality for both positive and negative values

An absolute value inequality can be turned into two separate inequalities: one for the positive case and one for the negative. So, we write two inequalities to represent the two scenarios: -2x + \(\frac{6}{7}\) > (\(12 - \frac{3}{7}\)) and -2x + \(\frac{6}{7}\) < - (\(12 - \frac{3}{7}\)). After simplifying, get: -2x > 11 and -2x < -12.

Step 3: Solve for x

We can continue solving these inequalities by dividing every part by -2. For the first inequality, -2x > 11; divide each side by -2 (which changes the direction of inequality): x < -5.5. For the second inequality, do the same: x > 6.

Step 4: Final Solution

The solution represents values of x for which original inequality holds. Therefore, we get the solution as two separate intervals: x < -5.5 , x > 6.