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a)
\(\left\{{}\begin{matrix}x^2+x+5< 0\\x^2-6x+1>0\end{matrix}\right.\)
\(\)Ta có
\(x^2+x+5=\left(x^2+x+\dfrac{1}{4}\right)+\dfrac{19}{4}=\left(x+\dfrac{1}{2}\right)^2+\dfrac{19}{4}\ge\dfrac{19}{4}>0\)
=> Bất phương trình đàu tiên sai, hệ bất phương trình sai
b)
\(\left\{{}\begin{matrix}2x^2+x-6>0\\3x^2-10x+3\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(2x-3\right)\left(x+2\right)>0\\\left(x-3\right)\left(3x-1\right)\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x>2\\x< -3\end{matrix}\right.\\\left[{}\begin{matrix}x\le-\dfrac{1}{3}\\x\ge3\end{matrix}\right.\end{matrix}\right.\)
c)
\(\left\{\begin{matrix} -x^2+4x-7< 0\\ x^2-2x-1\geq 0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x^2-4x+7>0\\ x^2-2x+1\geq 2\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} (x-2)^2+3>0\\ (x-1)^2-2\geq 0\end{matrix}\right.\Leftrightarrow (x-1)^2-2\geq 0\Leftrightarrow \left[\begin{matrix} x-1\geq \sqrt{2}\\ x-1\leq -\sqrt{2}\end{matrix}\right.\)
\(\Leftrightarrow \left[\begin{matrix} x\geq \sqrt{2}+1\\ x\leq 1-\sqrt{2}\end{matrix}\right.\)
d)
\(\left\{\begin{matrix} -2x^2-5x+4< 0\\ -x^2-3x+10>0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} 2x^2+5x-4>0\\ (2-x)(x+5)>0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} 2(x+\frac{5}{4})^2-\frac{57}{8}>0\\ (2-x)(x+5)>0\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} (x+\frac{5}{4}-\frac{\sqrt{57}}{4})(x+\frac{5}{4}+\frac{\sqrt{57}}{4})>0\\ (2-x)(x+5)>0\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} \left[\begin{matrix} x>\frac{-5+\sqrt{57}}{4}\\ x< \frac{-5-\sqrt{57}}{4}\end{matrix}\right.\\ -5< x< 2\end{matrix}\right.\) \(\Rightarrow \left[\begin{matrix} -5< x< \frac{-5-\sqrt{57}}{4}\\ \frac{\sqrt{57}-5}{4}< x< 2\end{matrix}\right.\)
a)
\(\left\{\begin{matrix} 2x^2+9x+7>0\\ x^2+x-6< 0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} (x+1)(2x+7)>0\\ (x-2)(x+3)< 0\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} \left[\begin{matrix} x>-1\\ x< \frac{-7}{2}\end{matrix}\right.\\ -3< x< 2\end{matrix}\right.\Rightarrow -1< x< 2\)
b) \(\left\{\begin{matrix} 2x^2+x-6>0\\ 3x^2-10x+3\geq 0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} (2x-3)(x+2)>0\\ (x-3)(3x-1)\geq 0\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} \left[\begin{matrix} x>\frac{3}{2}\\ x< -2\end{matrix}\right.\\ \left[\begin{matrix} x\geq 3\\ x\leq \frac{1}{3}\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow \left[\begin{matrix} x\geq 3\\ x< -2\end{matrix}\right.\)
a: Đặt |x-6|=a, |y+1|=b
Theo đề, ta có hệ phương trình:
\(\left\{{}\begin{matrix}2a+3b=5\\5a-4b=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=1\end{matrix}\right.\)
=>|x-6|=1 và |y+1|=1
\(\Leftrightarrow\left\{{}\begin{matrix}x\in\left\{7;5\right\}\\y\in\left\{0;-2\right\}\end{matrix}\right.\)
b: Đặt |x+y|=a, |x-y|=b
Theo đề, ta có: \(\left\{{}\begin{matrix}2a-b=19\\3a+2b=17\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=\dfrac{55}{7}\\b=-\dfrac{23}{7}\left(loại\right)\end{matrix}\right.\)
=>HPTVN
c: Đặt |x+y|=a, |x-y|=b
Theo đề ta có: \(\left\{{}\begin{matrix}4a+3b=8\\3a-5b=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=2\\b=0\end{matrix}\right.\)
=>|x+y|=2 và x=y
=>|2x|=2 và x=y
=>x=y=1 hoặc x=y=-1
a: =>(x+1)(x+5)>0 và (x+3)(x-2)<0
=>\(\left\{{}\begin{matrix}x\in\left(-\infty;-5\right)\cup\left(-1;+\infty\right)\\x\in\left(-3;2\right)\end{matrix}\right.\Leftrightarrow x\in\left(-1;2\right)\)
b: =>(x+2)(2x-3)>0và 3x^2-10x+3>=0
=>(x+2)(2x-3)>0 và (x-3)(3x-1)>=0
=>\(\left\{{}\begin{matrix}x\in\left(-\infty;-2\right)\cup\left(\dfrac{3}{2};+\infty\right)\\x\in(-\infty;\dfrac{1}{3}]\cup[3;+\infty)\end{matrix}\right.\Leftrightarrow x\in\left(-\infty;-2\right)\cup[3;+\infty)\)