Solve |5x - 7| > 2x + 1: Absolute Value Inequality Challenge

Question

Given:

$\left|5x - 7\right| > 2x + 1$

Which of the following statements is necessarily true?

To solve the inequality $\left|5x - 7\right| > 2x + 1$ , we consider two scenarios for the absolute value expression.

Case 1: $5x - 7 > 2x + 1$

Rearrange terms:

$5x - 2x > 1 + 7$

$3x > 8$

$x > \frac{8}{3}$

Case 2: $-(5x - 7) > 2x + 1$

This simplifies to:

$-5x + 7 > 2x + 1$

Rearrange terms:

$7 - 1 > 2x + 5x$

$6 > 7x$

$x < \frac{6}{7}$

Hence, the solution combines both cases as:

$x < \frac{6}{7} \text{ or } x > \frac{8}{3}$

$x < \frac{6}{7} \text{ or } x > \frac{8}{3}$