To solve the inequality \(2 - 5x > 2|x - 3|\), we consider the expression inside the absolute value, \(|x - 3|\), which can be split into two cases:
Case 1: \(x - 3 \geq 0\) (i.e., \(x \geq 3\))
In this case, \(|x - 3| = x - 3\). The inequality becomes:
\(2 - 5x > 2(x - 3)\)
\(2 - 5x > 2x - 6\)
\(2 + 6 > 2x + 5x\)
\(8 > 7x\)
\(x < \frac{8}{7}\)
However, since \(x \geq 3\), there is no solution in this case.
Case 2: \(x - 3 < 0\) (i.e., \(x < 3\))
In this case, \(|x - 3| = -(x - 3) = -x + 3\). The inequality becomes:
\(2 - 5x > 2(-x + 3)\)
\(2 - 5x > -2x + 6\)
\(2 - 6 > -2x + 5x\)
\(-4 > 3x\)
\(x < -\frac{4}{3}\)
Thus, the solution is \(x < -\frac{4}{3}\).
