Answer:
[tex]x>\dfrac{3}{11}[/tex]
Step-by-step explanation:
If a, b and c are three sides of a triangle, then
[tex]a+b>c\\ \\a+c>b\\ \\b+c>a[/tex]
In your case,
[tex]a=3x+4\\ \\b=6x-1\\ \\c=8x+2[/tex]
Thus,
1.
[tex]3x+4+6x-1>8x+2\\ \\9x+3>8x+2\\ \\9x-8x>2-3\\ \\x>-1[/tex]
2.
[tex]3x+4+8x+2>6x-1\\ \\11x+6>6x-1\\ \\11x-6x>-1-6\\ \\5x>-7\\ \\x>-1.4[/tex]
3.
[tex]6x-1+8x+2>3x+4\\ \\14x+1>3x+4\\ \\14x-3x>4-1\\ \\11x>3\\ \\x>\dfrac{3}{11}[/tex]
Thus,
[tex]x>\dfrac{3}{11}[/tex]
Note that a>0, b>0 and c>0, so
1.
[tex]3x+4>0\\ \\3x>-4\\ \\x>-\dfrac{4}{3}[/tex]
2.
[tex]6x-1>0\\ \\6x>1\\ \\x>\dfrac{1}{6}[/tex]
3.
[tex]8x+2>0\\ \\8x>-2\\ \\x>-\dfrac{1}{4}[/tex]
Thus,
[tex]x>\dfrac{1}{6}[/tex]
As result,
[tex]x>\dfrac{3}{11}[/tex]
