To answer this question, we will consider the following cases:
1.-
[tex]x\leq0.[/tex]
If
[tex]\begin{gathered} x\ne-3, \\ |2x+6|>0>3x. \end{gathered}[/tex]
If
[tex]\begin{gathered} x=-3, \\ |2*-3+6|=0>3*-3. \end{gathered}[/tex]
Case 2.
[tex]x>0.[/tex]
Recall that:
[tex]|a|>b,\text{ if and only if }a>b\text{ or -a}>b.[/tex]
Therefore:
[tex]2x+6>3x\text{ or -}2x-6>3x.[/tex]
Solving each inequality for x, we get:
[tex]\begin{gathered} 2x+6>3x, \\ 6>x, \\ -2x-6>3x, \\ -5x>6, \\ x>-\frac{6}{5}. \end{gathered}[/tex]
The union of the solutions in the number line is represented as follows:
Answer: None of the above.
