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

Because absolute value measures distance from zero! When $|5x - 7| > 2x + 1$, either 5x - 7 is positive and greater than 2x + 1, OR 5x - 7 is negative and its opposite is greater than 2x + 1.

For absolute value inequalities with 'greater than' (>), use 'or' because values can satisfy either case. For 'less than' (<), you typically use 'and' because values must be between the boundaries.

That's normal! Sometimes one case yields no solution or contradicts the other. Simply ignore the impossible case and use only the valid solutions in your final answer.

Pick a test value from each interval and substitute into the original inequality. For example, try x = 0 (from $x < \frac{6}{7}$) and x = 3 (from $x > \frac{8}{3}$) to verify they work!

Because the values that make the inequality true are not continuous! There's a gap between $\frac{6}{7}$ and $\frac{8}{3}$ where the inequality is false, so we write the solution as two separate regions.