To solve the inequality $|6 - 2x| > 1$, we proceed as follows:

Step 1: Break the absolute value inequality

For any absolute value inequality of the form $|A| > B$, where $B > 0$, it is equivalent to: $A < -B \quad \text{or} \quad A > B$ Here, $A = 6 - 2x$ and $B = 1$, so: $6 - 2x < -1 \quad \text{or} \quad 6 - 2x > 1$

Step 2: Solve each inequality separately

Inequality 1: $6 - 2x < -1$

$6 - 2x < -1$ $-2x < -1 - 6$ $-2x < -7$ Divide by $-2$ (remember to reverse the inequality sign): $x > \frac{7}{2}$

Inequality 2: $6 - 2x > 1$

$6 - 2x > 1$ $-2x > 1 - 6$ $-2x > -5$ Divide by $-2$ (again, reverse the inequality sign): $x < \frac{5}{2}$

Step 3: Combine the results

The solution is: $x > \frac{7}{2} \quad \text{or} \quad x < \frac{5}{2}$

In interval notation, this is: $(-\infty, \frac{5}{2}) \cup (\frac{7}{2}, \infty)$

Final Answer:

$x \in (-\infty, \frac{5}{2}) \cup (\frac{7}{2}, \infty)$

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Would you like a detailed graph of this solution or further clarification?
Here are 5 related questions you might find helpful:

1. How do we solve $|A| < B$ inequalities?
2. What happens if the inequality is $|A| \geq B$?
3. Can we graph $|6 - 2x| > 1$ and interpret the solution visually?
4. How does reversing the inequality sign work when dividing by a negative number?
5. What are some real-life applications of absolute value inequalities?

Tip: Always check your solution intervals by testing points from each region!