Solve the inequality.

StartAbsoluteValue StartFraction 2 x minus 6 Over 3 EndFraction EndAbsoluteValue greater than or equals five sixths2x−63≥56

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Part 1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

We are tasked with solving the inequality:

\[
\left| \frac{2x - 6}{3} \right| \geq \frac{5}{6}
\]

### Step 1: Split the absolute value inequality
The absolute value inequality \( |A| \geq B \) means:
\[
A \geq B \quad \text{or} \quad A \leq -B
\]
So, we can split the inequality as:

\[
\frac{2x - 6}{3} \geq \frac{5}{6} \quad \text{or} \quad \frac{2x - 6}{3} \leq -\frac{5}{6}
\]

### Step 2: Solve each inequality

#### Inequality 1: 
\[
\frac{2x - 6}{3} \geq \frac{5}{6}
\]
Multiply both sides by 6 to eliminate the fractions:
\[
6 \times \frac{2x - 6}{3} \geq 6 \times \frac{5}{6}
\]
This simplifies to:
\[
2(2x - 6) \geq 5
\]
Distribute the 2:
\[
4x - 12 \geq 5
\]
Add 12 to both sides:
\[
4x \geq 17
\]
Divide by 4:
\[
x \geq \frac{17}{4}
\]

#### Inequality 2:
\[
\frac{2x - 6}{3} \leq -\frac{5}{6}
\]
Multiply both sides by 6:
\[
6 \times \frac{2x - 6}{3} \leq 6 \times -\frac{5}{6}
\]
This simplifies to:
\[
2(2x - 6) \leq -5
\]
Distribute the 2:
\[
4x - 12 \leq -5
\]
Add 12 to both sides:
\[
4x \leq 7
\]
Divide by 4:
\[
x \leq \frac{7}{4}
\]

### Step 3: Combine the results
From the two inequalities, we have:
\[
x \geq \frac{17}{4} \quad \text{or} \quad x \leq \frac{7}{4}
\]

Thus, the solution is:
\[
x \in \left( -\infty, \frac{7}{4} \right] \cup \left[ \frac{17}{4}, \infty \right)
\]

Would you like further details on this? Here are 5 related questions:
1. What happens if the inequality was \( < \) instead of \( \geq \)?
2. How do we solve an absolute value equation instead of an inequality?
3. How does multiplying or dividing by a negative number affect inequalities?
4. What if the inequality involved a different fraction inside the absolute value?
5. Can absolute value inequalities have no solutions?

**Tip:** When working with absolute value inequalities, always consider the two cases (positive and negative) carefully.

Solve the inequality. StartAbsoluteValue StartFraction 2 x minus 6 Over 3 EndFraction EndAbsoluteValue greater than or equals five sixths2x−63≥56

Learn how to solve the absolute value inequality |(2x - 6)/3| ≥ 5/6 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the properties of absolute values and solving linear inequalities. Suitable for students in Grades 9-12.

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