How do I solve the following absolute value inequality and inequality problems properly?
1) $\newcommand\abs[1]{|#1|}\abs{2x+9}>x$
Solving this problem algebraically, I get
When $x > 0, x > -9$
Steps:
$2x + 9 > x$
$x > -9$
When $x < 0, x < -3$
Steps:
$2x + 9 < -x$
$x < -3$
However, the complete answer is all real numbers. What are the steps that I need to take to get the complete answer?
2) $15x^2 - 2 > x$
My solution: $15x^2 - 2 > x$
$15x^2 - 2 - x > 0$
$(3x+1)(5x-2) > 0$
$(3x+1) > 0$ and $(5x-2) > 0$
or
$(3x+1) < 0$ and $(5x-2) < 0$
Case: $(3x+1) > 0$ and $(5x-2) > 0$
Solution: $x > -1/3$ and $x > 2/5$
Case: $(3x+1) < 0$ and $(5x-2) < 0$
Solution: $x < -1/3$ and $x < 2/5$
Final Solution: $x > -1/3$ and $x > 2/5$ or $x < -1/3$ and $x < 2/5$
However, the answer to this problem is: $x > 2/5$ or $x < -1/3$