In Exercises 59–94, solve each absolute value inequality. $12 < \(\left\)| -2x + \(\frac{6}{7}\) \(\right\)| + \(\frac{3}{7}\)$

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1. Start by isolating the absolute value expression on one side of the inequality. Subtract \( \frac{3}{7} \) from both sides to get: \[ 12 - \frac{3}{7} < \left| -2x + \frac{6}{7} \right| \]
2. Simplify the left side by finding a common denominator and performing the subtraction: \[ \text{Calculate } 12 - \frac{3}{7} = \frac{84}{7} - \frac{3}{7} = \frac{81}{7} \]. So the inequality becomes: \[ \frac{81}{7} < \left| -2x + \frac{6}{7} \right| \]
3. Rewrite the inequality to the standard form for absolute value inequalities: \[ \left| -2x + \frac{6}{7} \right| > \frac{81}{7} \]
4. Recall that for \( |A| > B \) where \( B > 0 \), the solution splits into two inequalities: \[ A < -B \quad \text{or} \quad A > B \]. Apply this to get: \[ -2x + \frac{6}{7} < -\frac{81}{7} \quad \text{or} \quad -2x + \frac{6}{7} > \frac{81}{7} \]
5. Solve each inequality separately for \( x \). For each, subtract \( \frac{6}{7} \) from both sides, then divide by \( -2 \), remembering to reverse the inequality sign when dividing by a negative number.

Key Concepts

Absolute Value Inequalities

An absolute value inequality involves expressions within absolute value bars and requires finding all values of the variable that satisfy the inequality. The absolute value represents the distance from zero, so inequalities often split into two cases to solve.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This often involves subtracting or adding constants and combining like terms to simplify the inequality for further analysis.

Solving Compound Inequalities

Absolute value inequalities can translate into compound inequalities, such as 'less than' or 'greater than' forms. Understanding how to split and solve these compound inequalities is essential to find the solution set for the variable.