Question

Solve and write the solution in interval notation: 3|7x-1|+2>17

Answer 1

To solve the inequality \( 3|7x - 1| + 2 > 17 \), we first isolate the absolute value expression.

1. Subtract 2 from both sides:
\[
3|7x - 1| > 15
\]

2. Divide both sides by 3:
\[
|7x - 1| > 5
\]

Next, we will solve the absolute value inequality \( |7x - 1| > 5 \).

This inequality can be split into two separate cases:

**Case 1:** \( 7x - 1 > 5 \)
\[
7x > 6 \quad \Rightarrow \quad x > \frac{6}{7}
\]

**Case 2:** \( 7x - 1 < -5 \)
\[
7x < -4 \quad \Rightarrow \quad x < -\frac{4}{7}
\]

Combining both cases, we find:
1. \( x > \frac{6}{7} \)
2. \( x < -\frac{4}{7} \)

In interval notation, the solution is:
\[
(-\infty, -\frac{4}{7}) \cup (\frac{6}{7}, \infty)
\]

Thus, the final solution in interval notation is:
\[
\boxed{(-\infty, -\frac{4}{7}) \cup (\frac{6}{7}, \infty)}
\]