# Chapter 5 - Relationships Within Triangles - Chapter Review - Page 344: 34

$1 < x < 25$

#### Work Step by Step

Using the Triangle Inequality Theorem, we need to see if the sum of each of the combinations of two sides is greater than the other side. Let's set $x$ as the length of the third side. Now, let's look at the possible combinations of sides to see if the lengths of two sides is greater than the length of a third side: 1st combination: $x + 12 > 13$ Solve for $x$ by subtracting $12$ from each side of the inequality: $x > 1$ 2nd combination: $x + 13 > 12$ Subtract $13$ from each side of the equation: $x > -1$ 3rd combination: $12 + 13 > x$ Switch the inequality around and add to simplify: $x < 25$ $x$ cannot be a negative number because it is a length, so let's eliminate the inequality that includes a negative number. $1 < x < 25$

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