Question

select the correct answer. which inequality represents the values of x that ensure triangle abc exists? (triangle abc with sides: ab = 2x + 4, bc = 6x, ac = 18) options: a. \\(\frac{7}{4} < x < \frac{11}{3}\\) b. \\(-\frac{7}{4} < x < \frac{11}{3}\\) c. \\(1 < x < 5\\) d. \\(2 < x < 6\\)

Explanation:

Step1: Apply Triangle Inequality (Sum > Third Side)

For triangle \(ABC\) with sides \(AB = 2x + 4\), \(BC = 6x\), and \(AC = 18\), we use the triangle inequality theorem: the sum of any two sides must be greater than the third side. We consider three inequalities:

1. \(AB + BC > AC\): \((2x + 4) + 6x > 18\)

Simplify: \(8x + 4 > 18\) → \(8x > 14\) → \(x > \frac{7}{4}\)

1. \(AB + AC > BC\): \((2x + 4) + 18 > 6x\)

Simplify: \(2x + 22 > 6x\) → \(22 > 4x\) → \(x < \frac{11}{2}\)

1. \(BC + AC > AB\): \(6x + 18 > 2x + 4\)

Simplify: \(4x > -14\) → \(x > -\frac{7}{2}\) (This is always true since \(x > \frac{7}{4}\) from step 1)

Step2: Combine Valid Inequalities

From step 1, the relevant inequalities are \(x > \frac{7}{4}\) and \(x < \frac{11}{2}\), so \(\frac{7}{4} < x < \frac{11}{2}\).

Answer:

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