Question

the side lengths of a triangle are 9, 12, and x units. (the figure above gives only one possible configuration for the triangle.) determine the possible values of x. write your answer as an inequality. use x only once in your inequality.

Explanation:

Step1: Apply the triangle inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
So, \(9 + 12>x\), \(9+x>12\), and \(12 + x>9\).

Step2: Simplify the inequalities

For \(9 + 12>x\), we get \(21>x\) or \(x < 21\).
For \(9+x>12\), subtract 9 from both sides: \(x>12 - 9\), so \(x>3\).
The inequality \(12 + x>9\) is always true for positive \(x\) (since \(x>0\) as it represents a side - length).

Answer:

\(3