<s>[INST] If three sides of an acute triangle is $3$ cm, $4$ cm, and $x$ cm. What are the possible values of $x$? Problem: If three sides of an acute triangle is 3 cm, 4 cm, and $x$ cm. What are the possible values of $x$?
2 of the choices: (A) $1<x<5$, and (B) $0<x<7$
Solution: According to triangle theorems, 2 sides of a triangle is greater than the 3rd side and their difference is less than the 3rd side. 
From one of the choices, (ranges $0<x<7$) when $x = 6$:


*

*$3 + 4 > 6$

*$6 + 4 > 3$

*$3 + 6 > 4$
From the above, it means $6$ is a possible value for $x$, and since choice A doesn't have $6$ in it, I chose B as the correct answer. However that isn't the case, the correct answer according to the book is A, which I don't know why. So how was it $1<x<5$? Any help would be appreciated. 
 [/INST] Let the angles opposite the sides of lengths $x,3,4$ be $\theta,\omega, \phi$ respectively. Then by the Cosine Rule we obtain the relations: $$x^2=3^2+4^2-2\cdot3\cdot4\cos \theta,$$ $$3^2=x^2+4^2-2\cdot x\cdot4\cos \omega$$ and $$4^2=x^2+3^2-2\cdot x\cdot3\cos \phi$$ for $0<\cos \theta, \cos \omega,\cos \phi<1$.
From the first relation we have that $x^2=25-24\cos\theta$, which tells us that the extremes of $x^2$ are $25-24\times 1=1$ and $25-24\times 0=25$, so that $1<x^2<25$, or $1<x<5$.
From the second relation we deduce that $x=4\cos\omega\pm\sqrt{16\cos^2\omega-7}$, from which we learn that $x=\sqrt{-7}$ when $\cos\omega=0$ and $x=1,7$ when $\cos\omega=1$. Any way we look at these, they tell us nothing interesting.
From the third relation we have that $x=3\cos\phi\pm\sqrt{9\cos^2\phi+7}$, which tells us that $x=\sqrt7$ when $\cos\phi=0$ and $x=-1,7$ when $\cos\phi=1$. We have the possible interesting relations $\sqrt7<x<7$.
Finally, combining the inequalities $1<x<5$ and $\sqrt 7<x<7$, we have the tighter bounds $\sqrt 7<x<5$.
PS. I have tacitly assumed throughout that the sinusoidal functions $x(\theta),x(\omega),x(\phi)$ which we analysed are either strictly increasing or strictly decreasing (indeed they're all strictly decreasing) wherever they are defined in the intervals $0<\theta,\omega,\phi<π/2$ respectively. This can however be easily justified if needed.
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