Find the range of possible measures of the third side. Use the variable \( x \). \[ 4,7 \]

To find the range of possible measures for the third side \( x \) of a triangle with sides 4 and 7, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. 1. For the sides 4 and \( x \): \[ 4 + x > 7 \quad \Rightarrow \quad x > 3 \] 2. For the sides 7 and \( x \): \[ 7 + x > 4 \quad \Rightarrow \quad x > -3 \] (but since \( x \) must be positive, this isn't a restriction) 3. For the sides 4 and 7: \[ 4 + 7 > x \quad \Rightarrow \quad x < 11 \] Combining these results, the range of \( x \) is: \[ 3 < x < 11 \] So, the possible measures for the third side \( x \) fall between 3 and 11.