Find the range of possible measures of the third side. $$ 8,17 $$

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Let the third side have measure $$ x $$. The triangle inequalities require that the sum of any two sides is greater than the third side. 1. From the sides $$ 8 $$ and $$ 17 $$: $$ 8 + 17 > x \quad \Longrightarrow \quad x < 25 $$ 2. From the sides $$ 8 $$ and $$ x $$: $$ 8 + x > 17 \quad \Longrightarrow \quad x > 9 $$ 3. From the sides $$ 17 $$ and $$ x $$: $$ 17 + x > 8 \quad \Longrightarrow \quad x > -9 $$ The inequality $$ x > -9 $$ is automatically satisfied when $$ x > 9 $$. Thus, the values of $$ x $$ that satisfy all the inequalities are those for which: $$ 9 < x < 25 $$ The third side must be greater than 9 and less than 25.

Find the range of possible measures of
the third side.

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