salmanhiro

OSN SMP 2017 - Hari Pertama No. 1

2 posts in this topic

Carilah semua bilangan real \(x\) yang memenuhi pertidaksamaan

\(\frac{x^{2}-3}{x^{2}-1} + \frac{x^{2}+5}{x^{2}+3}\geq \frac{x^{2}-5}{x^{2}-3}+\frac{x^{2}+3}{x^{2}+1}\)

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\begin{align*}\frac{x^{2}-3}{x^{2}-1}+\frac{x^{2}+5}{x^{2}+3}&\geq\frac{x^{2}-5}{x^{2}-3}+\frac{x^{2}+3}{x^{2}+1}\\\frac{x^{2}-1-2}{x^{2}-1}+\frac{x^{2}+3+2}{x^{2}+3}&\geq\frac{x^{2}-3-2}{x^{2}-3}+\frac{x^{2}+1+2}{x^{2}+1}\\1-\frac{2}{x^2-1}+1+\frac{2}{x^2+3}&\geq 1-\frac{2}{x^2-3}+1+\frac{2}{x^2+1}\\-\frac{1}{x^2-1}+\frac{1}{x^2+3}&\geq -\frac{1}{x^2-3}+\frac{1}{x^2+1}\\\frac{1}{x^2-3}+\frac{1}{x^2+3}+\frac{-1}{x^2-1}+\frac{-1}{x^2+1}&\geq 0\\\frac{2x^2}{(x^2-3)(x^2+3)}-\frac{2x^2}{(x^2-1)(x^2+1)}&\geq 0\\\frac{x^2(x^4-1)-x^2(x^4-9)}{(x^2-3)(x^2-1)(x^2+1)(x^2+3)}&\geq 0,\ x^2+1\ \text{dan}\ x^2+3\ \text{definit positif}\\\frac{x^2}{(x+\sqrt{3})(x-\sqrt{3})(x-1)(x+1)}&\geq 0.\end{align*} Dengan menggunakan garis bilangan, maka didapat solusinya $x\in(-\infty, -\sqrt{3})\cup(-1, 1)\cup(\sqrt{3}, \infty)$.

 

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