b.
Với \(x=0\) không phải nghiệm
Với \(x\ne0\) hệ tương đương:
\(\left\{{}\begin{matrix}\dfrac{y}{x^2}+\dfrac{y^2}{x}=-6\\\dfrac{1}{x^3}+y^3=19\end{matrix}\right.\)
Đặt \(\left(\dfrac{1}{x};y\right)=\left(u;v\right)\) ta được: \(\left\{{}\begin{matrix}uv^2+u^2v=-6\\u^3+v^3=19\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}3uv^2+3u^2v=-18\\u^3+v^3+19\end{matrix}\right.\)
Cộng vế với vế:
\(\left(u+v\right)^3=1\Rightarrow u+v=1\)
Thay vào \(u^2v+uv^2=-6\Rightarrow uv=-6\)
Theo Viet đảo, u và v là nghiệm của:
\(t^2-t-6=0\) \(\Rightarrow\left[{}\begin{matrix}t=-2\\t=3\end{matrix}\right.\) \(\Rightarrow\left(u;v\right)=\left(-2;3\right);\left(3;-2\right)\)
\(\Rightarrow\left(\dfrac{1}{x};y\right)=\left(-2;3\right);\left(3;-2\right)\)
\(\Rightarrow\left(x;y\right)=\left(-\dfrac{1}{2};3\right);\left(\dfrac{1}{3};-2\right)\)
a.
ĐKXĐ: \(x\ne3\)
- Với \(x\ge0\) pt trở thành:
\(\dfrac{x^2-x-12}{x-3}=2x\Rightarrow x^2-x-12=2x^2-6x\)
\(\Leftrightarrow x^2-5x+12=0\) (vô nghiệm)
- Với \(x< 0\) pt trở thành:
\(\dfrac{x^2+x-12}{x-3}=2x\Rightarrow\dfrac{\left(x-3\right)\left(x+4\right)}{x-3}=2x\)
\(\Rightarrow x+4=2x\Rightarrow x=4>0\) (ktm)
Vậy pt đã cho vô nghiệm
e: \(\begin{cases}x\left(x+5\right)<4x+2\\ \left(2x-1\right)\left(x+3\right)\ge4x\end{cases}\Rightarrow\begin{cases}x^2+5x-4x-2<0\\ 2x^2+6x-x-3-4x\ge0\end{cases}\)
=>\(\begin{cases}x^2+x-2<0\\ 2x^2+x-3\ge0\end{cases}\Rightarrow\begin{cases}\left(x+2\right)\left(x-1\right)<0\\ 2x^2+3x-2x-3\ge0\end{cases}\)
=>\(\begin{cases}-2
=>-2<x<=-1
f: ĐKXĐ: x∉{1;4;2;5}
Ta có: \(\frac{1}{x^2-5x+4}\le\frac{1}{x^2-7x+10}\)
=>\(\frac{1}{x^2-5x+4}-\frac{1}{x^2-7x+10}\le0\)
=>\(\frac{x^2-7x+10-x^2+5x-4}{\left(x-1\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)}\le0\)
=>\(\frac{-2x+6}{\left(x-1\right)\left(x-4\right)\left(x-2\right)\left(x-5\right)}\le0\)
=>\(\frac{x-3}{\left(x-1\right)\left(x-2\right)\left(x-4\right)\left(x-5\right)}\ge0\)
Đặt \(A=\frac{x-3}{\left(x-1\right)\left(x-2\right)\left(x-4\right)\left(x-5\right)}\)
Đặt x-3=0
=>x=3
Đặt x-1=0
=>x=1
Đặt x-2=0
=>x=2
Đặt x-4=0
=>x=4
Đặt x-5=0
=>x=5
Bảng xét dấu:
Theo bãng xét dấu, ta có: A>=0 khi 1<x<2; 3<=x<4; x>5
a, ĐKXĐ : \(D=R\)
BPT \(\Leftrightarrow x^2+5x+4< 5\sqrt{x^2+5x+4+24}\)
Đặt \(x^2+5x+4=a\left(a\ge-\dfrac{9}{4}\right)\)
BPTTT : \(5\sqrt{a+24}>a\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a+24\ge0\\a< 0\end{matrix}\right.\\\left\{{}\begin{matrix}a\ge0\\25\left(a+24\right)>a^2\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}-24\le a< 0\\\left\{{}\begin{matrix}a^2-25a-600< 0\\a\ge0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}-24\le a< 0\\0\le a< 40\end{matrix}\right.\)
\(\Leftrightarrow-24\le a< 40\)
- Thay lại a vào ta được : \(\left\{{}\begin{matrix}x^2+5x-36< 0\\x^2+5x+28\ge0\end{matrix}\right.\)
\(\Leftrightarrow-9< x< 4\)
Vậy ....
b, ĐKXĐ : \(x>0\)
BĐT \(\Leftrightarrow2\left(\sqrt{x}+\dfrac{1}{2\sqrt{x}}\right)< x+\dfrac{1}{4x}+1\)
- Đặt \(\sqrt{x}+\dfrac{1}{2\sqrt{x}}=a\left(a\ge\sqrt{2}\right)\)
\(\Leftrightarrow a^2=x+\dfrac{1}{4x}+1\)
BPTTT : \(2a\le a^2\)
\(\Leftrightarrow\left[{}\begin{matrix}a\le0\\a\ge2\end{matrix}\right.\)
\(\Leftrightarrow a\ge2\)
\(\Leftrightarrow a^2\ge4\)
- Thay a vào lại BPT ta được : \(x+\dfrac{1}{4x}-3\ge0\)
\(\Leftrightarrow4x^2-12x+1\ge0\)
\(\Leftrightarrow x=(0;\dfrac{3-2\sqrt{2}}{2}]\cup[\dfrac{3+2\sqrt{2}}{2};+\infty)\)
Vậy ...
a, hệ\(\Leftrightarrow\)$\left \{ {{x>\frac{1}{2} } \atop {x<m+2}} \right.$
để hệ có nghiệm ⇒ m+2< $\frac{1}{2}$ ⇒ m<$\frac{-3}{2}$
a, ĐKXĐ : \(\left[{}\begin{matrix}x\le-3\\x\ge0\end{matrix}\right.\)
TH1 : \(x\le-3\) ( LĐ )
TH2 : \(x\ge0\)
BPT \(\Leftrightarrow x^2+2x+x^2+3x+2\sqrt{\left(x^2+2x\right)\left(x^2+3x\right)}\ge4x^2\)
\(\Leftrightarrow\sqrt{\left(x^2+2x\right)\left(x^2+3x\right)}\ge x^2-\dfrac{5}{2}x\)
\(\Leftrightarrow2\sqrt{\left(x+2\right)\left(x+3\right)}\ge2x-5\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x< \dfrac{5}{2}\\x\ge-2\end{matrix}\right.\\\left\{{}\begin{matrix}x\ge\dfrac{5}{2}\\4x^2+20x+24\ge4x^2-20x+25\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}0\le x< \dfrac{5}{2}\\x\ge\dfrac{5}{2}\end{matrix}\right.\)
\(\Leftrightarrow x\ge0\)
Vậy \(S=R/\left(-3;0\right)\)
a)\(\left\{{}\begin{matrix}3x+2\ge12-2x\\-4x^2+12x-5\le0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}5x\ge10\\-4x^2+12x-5\le0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x\ge2\\\left[{}\begin{matrix}\dfrac{1}{2}\ge x\\x\ge\dfrac{5}{2}\end{matrix}\right.\end{matrix}\right.\Leftrightarrow x\ge\dfrac{5}{2}\)
a)\(\left\{{}\begin{matrix}3x+2\ge12-2x\\-4x^2+12x-5\le0\end{matrix}\right.\)⇔\(\left\{{}\begin{matrix}5x\ge10\\-4x^2+12x-5\le0\end{matrix}\right.\)⇔\(\left\{{}\begin{matrix}x\ge2\\\left[{}\begin{matrix}\dfrac{1}{2}\ge x\\x\ge\dfrac{5}{2}\end{matrix}\right.\end{matrix}\right.\)⇔ x\(\ge\)\(\dfrac{5}{2}\)
b) Ta có -x^2 + 3x - 4 < 0−x2+3x−4<0 \forall x \in \mathbb{R}∀x∈R.
\(\dfrac{\text{x^2 −mx−2}}{-x^2+3x-4}< 1\)⇔x2−mx−2>−x2+3x−4
⇔2x2−(m+3)x+2>0∀x∈R⇔\(\left\{{}\begin{matrix}a>0\\\text{Δ< 0 }\Delta< 0\end{matrix}\right.\)⇔\(^{^{^{ }}m^2}\)
+6m−7<0⇔−7<m<1.
c) Điều kiện x>0x>0
Phương trình \Leftrightarrow⇔ \(\sqrt{x^2+1+\dfrac{1}{x^2}}+\sqrt{x-1+\dfrac{1}{x}}\le\sqrt{\left(x+\dfrac{1}{x}\right)^3}\) ⇔\(\sqrt{\left(x+\dfrac{1}{x}\right)^2-1}+\sqrt{x-1+\dfrac{1}{x}}\le\sqrt{\left(x+\dfrac{1}{x}\right)}\) ⇔\(\sqrt{\left(x+\dfrac{1}{x}+1\right)\left(x+\dfrac{1}{x}-1\right)}+\sqrt{x-1+\dfrac{1}{x}}\le\sqrt{\left(x+\dfrac{1}{x}\right)^3}\) *
Phương trình (*)(∗) trở thành:
(t+1)(t−1)+t−1≤t3⇔t−1(t+1+1)
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