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 when the other two sides are given as \( 4 \) and \( 7 \), we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given the sides \( a = 4 \) and \( b = 7 \), we can set up the following inequalities based on the triangle inequality: 1. \( a + b > x \) 2. \( a + x > b \) 3. \( b + x > a \) Now, let's substitute the values of \( a \) and \( b \) into these inequalities: 1. \( 4 + 7 > x \) 2. \( 4 + x > 7 \) 3. \( 7 + x > 4 \) Now, we will simplify each inequality: 1. \( 11 > x \) or \( x < 11 \) 2. \( x > 3 \) 3. \( x > -3 \) (This inequality is always true since \( x \) must be positive) From the first two inequalities, we can combine them to find the range of \( x \): - From \( x < 11 \) - From \( x > 3 \) Thus, the range of possible measures for the third side \( x \) is: \[ 3 < x < 11 \] This means that \( x \) can take any value greater than \( 3 \) and less than \( 11 \).