https://file.aiquickdraw.com/mathbot/file/17271073580673du4bk65.jpg

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!

Solve the absolute value inequality |(2/3)x + 7/6| ≥ 1/4

Learn how to solve the absolute value inequality |(2/3)x + 7/6| ≥ 1/4 using algebraic techniques. This problem covers core concepts such as splitting absolute value inequalities into two cases and solving them step-by-step. Suitable for students in Grades 8-10.

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