To determine the range of values for the third side \( x \) of a triangle when the other two sides are given as 7 each, we can apply the triangle inequality theorem. This theorem states that for any triangle with sides \( a \), \( b \), and \( c \), the following inequalities must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
In this case, let the two known sides \( a \) and \( b \) both be 7, and let \( c \) be the unknown side \( x \). We can set up the inequalities as follows:
- \( 7 + 7 > x \) → \( 14 > x \) → \( x < 14 \)
- \( 7 + x > 7 \) → \( x > 0 \)
- \( 7 + x > 7 \) → \( x > 0 \) (This inequality does not provide new information.)
Now, combining the inequalities from the triangle inequality theorem gives us:
- \( x < 14 \)
- \( x > 0 \)
Thus, the range of values for the third side \( x \) is:
\[ 0 < x < 14 \]
Hence, \( x \) must be a positive number less than 14.
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