Question

1. the figure shows two triangles, where x represents the length of the common side. write an inequality that shows all the possible values of x.

Explanation:

Step1: Apply triangle - inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), the following three inequalities must hold: \(a + b>c\), \(a + c>b\), and \(b + c>a\).
Let's consider the two - triangle situation. Suppose the side lengths of the triangles are related to \(x\).
If we assume the side - length relationships in the triangles formed, for a triangle with side lengths \(x\), \(9\), and \(18\), we have \(9 + 18>x\) (i.e., \(x<27\)), \(9 + x>18\) (i.e., \(x > 9\)), and \(18+x>9\) (which is always true for non - negative \(x\)).
For a triangle with side lengths \(x\), \(13\), and \(10\), we have \(10 + 13>x\) (i.e., \(x<23\)), \(10 + x>13\) (i.e., \(x>3\)), and \(13 + x>10\) (which is always true for non - negative \(x\)).

Step2: Find the intersection of inequalities

To satisfy all triangle - inequality conditions for both triangles simultaneously, we need to find the intersection of the inequalities.
The inequalities from the first triangle are \(9 < x<27\), and from the second triangle are \(3 < x<23\).
The intersection of these two sets of inequalities is \(9 < x<23\).

Answer:

\(9 < x<23\)