Why did the text only check the case in which the sum of the lengths of the two shortest sides is jot greater than the length of the longest side?

Question

Teri has sticks of length 9 inches, 12 inches, and 14 inches. She cuts the same amount off of each stick. After cutting this amount off each stick, she is no longer able to make a triangle by attaching the sticks end to end. What is the smallest amount Teri could have cut off of each stick? This is the solution I found on the textbook: The only way Teri cannot make a triangle is if the sum of the lengths of the two shortest sticks is not greater than the length of the longest stick. If she cuts $x$ off each stick, then the sticks have lengths $9-x$, $12-x$, and $14-x$. The sum of the two shortest sticks then is $(9-x) + (12-x)$, which equals $21-2x$. If she can’t make a triangle, then this sum must be less than or equal to the length of the longest stick, so $21-2x \le 14-x$. Adding $2x$ to both sides and subtracting 14 from both sides gives $\le x$. Therefore, the smallest amount Teri could have cut off each stick is $\boxed{7\text{ inches}}$. If she cuts 7 inches off of each stick, she’ll have sticks with lengths 2 inches, 5 inches, and 7 inches. Since $2+5$ is not greater than 7, she cannot make a triangle with these sticks. Why did the text only check the case in which the sum of the lengths of the two shortest sticks is not greater than the length of the longest stick?

Answer 1

If the three numbers are $a,b,c$ with $a \le b \le c$ the text is checking the triangle inequality $a+b \gt c$. There is no need to check whether $a+c \gt b$ or $b+c \gt a$ because they are automatically true with $c$ on the left.