Question

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Answer 1

To determine the range of values for the third side of a triangle, we can apply the triangle inequality theorem, which states that for any triangle with sides \( a \), \( b \), and \( c \):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In your case, you're looking to find \( x \) as one of the triangle sides. Assuming you have the lengths of the other two sides, let's denote them as \( a \) and \( b \).

The inequalities we will use are:

1. \( a + b > x \) → This can be rearranged to \( x < a + b \)
2. \( a + x > b \) → This can be rearranged to \( x > b - a \) (if \( b > a \)) or \( x > a - b \) (if \( a > b \))
3. \( b + x > a \) → This can be rearranged to \( x > a - b \) (if \( a > b \)) or \( x > b - a \) (if \( b > a \))

By compiling these inequalities, we get the range:

\[
\max(a - b, b - a) < x < a + b
\]

If you provide the specific values for \( a \) and \( b \), I can give the precise range for \( x \). Otherwise, the answer remains generally as stated above.