Answer:
[tex]x > \frac{1}{2}[/tex]
Step-by-step explanation:
For the first one, in any triangle, we have the sum of two sides is greater than the third side. Hence, we have 3 inequalities: [tex]4x+3>2x+1, 5x+1>x+3, 3x+4>3x[/tex], which give us [tex]x>-1, x>\frac{1}{2},4>0[/tex]. We conclude that [tex]x>\frac{1}{2}[/tex].
For the second question, we have [tex]AB<BP+AP=2x[/tex], and similarly we can show that [tex]AC, BC<2x[/tex]. Hence, [tex]AB+BC+AC<6x[/tex].
The third question is not very clear.
