Let's solve the absolute value inequality provided in the image:

$\left|\frac{2}{3}x + \frac{7}{6}\right| \geq \frac{1}{4}$

Step 1: Split the inequality into two cases

The absolute value inequality $|A| \geq B$ can be written as two inequalities: $A \geq B \quad \text{or} \quad A \leq -B$

Thus, for this problem: $\frac{2}{3}x + \frac{7}{6} \geq \frac{1}{4} \quad \text{or} \quad \frac{2}{3}x + \frac{7}{6} \leq -\frac{1}{4}$

Step 2: Solve each inequality

Case 1: Solve $\frac{2}{3}x + \frac{7}{6} \geq \frac{1}{4}$

First, subtract $\frac{7}{6}$ from both sides: $\frac{2}{3}x \geq \frac{1}{4} - \frac{7}{6}$

We need a common denominator to subtract the fractions. The least common denominator between 4 and 6 is 12. Rewriting the fractions: $\frac{1}{4} = \frac{3}{12}, \quad \frac{7}{6} = \frac{14}{12}$ Now, subtract: $\frac{2}{3}x \geq \frac{3}{12} - \frac{14}{12} = -\frac{11}{12}$

Next, multiply both sides by $\frac{3}{2}$ to solve for $x$: $x \geq \left(-\frac{11}{12}\right) \times \frac{3}{2} = -\frac{33}{24} = -\frac{11}{8}$

Case 2: Solve $\frac{2}{3}x + \frac{7}{6} \leq -\frac{1}{4}$

Subtract $\frac{7}{6}$ from both sides: $\frac{2}{3}x \leq -\frac{1}{4} - \frac{7}{6}$

Again, find a common denominator for the fractions: $-\frac{1}{4} = -\frac{3}{12}, \quad \frac{7}{6} = \frac{14}{12}$ Now, subtract: $\frac{2}{3}x \leq -\frac{3}{12} - \frac{14}{12} = -\frac{17}{12}$

Multiply both sides by $\frac{3}{2}$: $x \leq \left(-\frac{17}{12}\right) \times \frac{3}{2} = -\frac{51}{24} = -\frac{17}{8}$

Step 3: Combine the two solutions

We have two inequalities: $x \geq -\frac{11}{8} \quad \text{or} \quad x \leq -\frac{17}{8}$

This represents a disjunction, meaning the solution is either $x \leq -\frac{17}{8}$ or $x \geq -\frac{11}{8}$.

Step 4: Graph the solution

From the graph options provided in the image, the graph that corresponds to these solutions is option D. The solution is represented as two intervals extending away from $-\frac{17}{8}$ and $-\frac{11}{8}$.

Would you like more details on any step? Here are some questions to consider:

1. What are some alternative methods to solve absolute value inequalities?
2. How do we determine when to use a conjunction (AND) versus a disjunction (OR)?
3. Why do we split the absolute value inequality into two cases?
4. How does multiplying or dividing by negative numbers affect inequality signs?
5. What would the solution look like if the inequality sign were reversed?

Tip: Always check if multiplying or dividing by negative numbers requires flipping the inequality sign!