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In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $15$. What is the greatest possible perimeter of the triangle? （2006 AMC10B, Question 10) 
Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities
Let $x$ be the length of the first side. The lengths of the sides are: $x, 3 x$, and 15 . By the Triangle Inequality, $$ \begin{aligned} \&3 x\textless x+15 \textbackslash\textbackslash{} \&2 x\textless15 \textbackslash\textbackslash{} \&x\textless\frac{15}{2} \end{aligned} $$ The greatest integer satisfying this inequality is 7 . So the greatest possible perimeter is $7+3 \cdot 7+15=$ (A) 43 