Question

select the correct answer.
which inequality represents the values of ( x ) that ensure triangle ( abc ) exists?
(image of triangle ( abc ) with sides: ( ab = 2x + 4 ), ( bc = 6x ), ( ac = 18 ))
a. ( \frac{7}{4} < x < \frac{11}{2} )
b. ( -\frac{7}{4} < x < \frac{11}{2} )
c. ( 1 < x < 5 )
d. ( 2 < x < 6 )

Explanation:

Step1: Apply Triangle Inequality Theorem

For a triangle with sides \(a\), \(b\), \(c\), the sum of any two sides must be greater than the third side. Let the sides be \(a = 2x + 4\), \(b = 6x\), \(c = 18\).

Inequality 1: \(a + b > c\)

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

Inequality 2: \(a + c > b\)

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

Inequality 3: \(b + c > a\)

\(6x + 18 > 2x + 4\)
\(4x > -14\)
\(x > -\frac{14}{4}=-\frac{7}{2}\) (But since side lengths must be positive, \(2x + 4>0\) and \(6x>0\) which gives \(x > - 2\) and \(x>0\). The most restrictive from the first two inequalities and positive side is \(x > \frac{7}{4}\) and \(x < \frac{11}{2}\))

Answer:

A. \(\frac{7}{4} < x < \frac{11}{2}\)