Question

select the correct answer.
which inequality represents the values of ( x ) that ensure triangle ( abc ) exists?

( \bigcirc a. \frac{7}{4} lt x lt \frac{11}{2} )
( \bigcirc b. -\frac{7}{4} lt x lt \frac{11}{2} )
( \bigcirc c. 1 lt x lt 5 )
( \bigcirc d. 2 lt x lt 6 )

Explanation:

Step1: Apply triangle - side - length relationship

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

1. \(AB + BC>AC\):

\((2x + 4)+6x>18\)
\(8x+4>18\)
\(8x>14\)
\(x>\frac{7}{4}\)

1. \(AB + AC>BC\):

\((2x + 4)+18>6x\)
\(2x+22>6x\)
\(22>4x\)
\(x < \frac{11}{2}\)

1. \(BC + AC>AB\):

\(6x+18>2x + 4\)
\(4x>-14\) (Since side - length \(x>0\) from the first two inequalities, this inequality is always satisfied for \(x>\frac{7}{4}\))

Answer:

A. \(\frac{7}{4}