Answer:
Option A.
Step-by-step explanation:
According to triangle inequality theorem, a triangle exists if the sum of two sides is greater that third side.
Apply this theorem on each combination.
Case I: [tex](2x+4)+6x>18[/tex]
[tex]8x+4>18[/tex]
[tex]8x>14[/tex]
[tex]x>\dfrac{14}{8}[/tex]
[tex]x>\dfrac{7}{4}[/tex] ...(1)
Case II: [tex](2x+4)+18>6x[/tex]
[tex]2x+22>6x[/tex]
[tex]22>4x[/tex]
[tex]\dfrac{22}{4}>x[/tex]
[tex]\dfrac{11}{2}>x[/tex] ...(2)
Case III: [tex]6x+18>2x+4[/tex]
[tex]6x-2x>4-18[/tex]
[tex]4x>-14[/tex]
[tex]x>-\dfrac{14}{4}[/tex]
[tex]x>-\dfrac{7}{2}[/tex] ...(3)
On combining (1), (2) and (3), we get
[tex]\dfrac{7}{4}<x<\dfrac{11}{2}[/tex]
Therefore, the correct option is A.
