4. Find the range of possible measures of the third side. Use the variable \( \mathbf{x} \). \[ 19,13 \]

To find the range of possible measures for the third side \( x \) of a triangle when the other two sides are given as \( 19 \) and \( 13 \), 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 \( a = 19 \) and \( b = 13 \), 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. \( 19 + 13 > x \) 2. \( 19 + x > 13 \) 3. \( 13 + x > 19 \) Now, we will simplify each inequality: 1. \( 32 > x \) or \( x < 32 \) 2. \( x > 13 - 19 \) or \( x > -6 \) (this inequality is always true since \( x \) must be positive) 3. \( x > 19 - 13 \) or \( x > 6 \) Now, combining the valid inequalities, we have: - From the first inequality: \( x < 32 \) - From the third inequality: \( x > 6 \) Thus, the range of possible measures for the third side \( x \) is: \[ 6 < x < 32 \] This means that the third side must be greater than \( 6 \) and less than \( 32 \).