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a: \(-3x^2\ge0\)
\(\Leftrightarrow x^2< =0\)
=>x=0
b: \(\dfrac{-5}{4x^2}\ge0\)
\(\Leftrightarrow4x^2< 0\)(vô lý)
c: \(\dfrac{4}{x+3}>=0\)
=>x+3>0
hay x>-3
d: \(\dfrac{-5}{2x-1}>=0\)
=>2x-1<0
hay x<1/2
e: \(\dfrac{-2}{x^2+1}>=0\)
=>x2+1<0(vô lý)
f: \(\dfrac{10}{x^2+9}>=0\)
=>x2+9>0(luôn đúng)
a: \(-3x^2\ge0\)
\(\Leftrightarrow x^2< =0\)
=>x=0
b: \(\dfrac{-5}{4x^2}\ge0\)
\(\Leftrightarrow4x^2< 0\)(vô lý)
c: \(\dfrac{4}{x+3}>=0\)
=>x+3>0
hay x>-3
d: \(\dfrac{-5}{2x-1}>=0\)
=>2x-1<0
hay x<1/2
e: \(\dfrac{-2}{x^2+1}>=0\)
=>x2+1<0(vô lý)
f: \(\dfrac{10}{x^2+9}>=0\)
=>x2+9>0(luôn đúng)
a/ \(2x^2-3x+1>0\Rightarrow\left[{}\begin{matrix}x>1\\x< \frac{1}{2}\end{matrix}\right.\)
b/ \(-3x^2+2x+1< 0\Rightarrow-\frac{1}{3}< x< 1\)
c/ \(\frac{x+3}{x-2}\ge0\Rightarrow\left[{}\begin{matrix}x>2\\x\le-3\end{matrix}\right.\)
d/ \(\frac{2x+1}{x+2}\ge1\Leftrightarrow\frac{2x+1}{x+2}-1\ge0\Leftrightarrow\frac{x-1}{x+2}\ge0\Rightarrow\left[{}\begin{matrix}x\ge1\\x< -2\end{matrix}\right.\)
e/ \(\frac{\sqrt{x}+3}{2-\sqrt{x}}\le0\Rightarrow\left\{{}\begin{matrix}x\ge0\\2-\sqrt{x}< 0\end{matrix}\right.\) \(\Rightarrow x>4\)
g/\(\frac{\sqrt{x}-3}{\sqrt{x}-2}\ge0\Rightarrow\left\{{}\begin{matrix}x\ge0\\\left[{}\begin{matrix}x\ge9\\x< 4\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x\ge0\\0\le x< 4\end{matrix}\right.\)
h/ \(\frac{\sqrt{x}-3}{\sqrt{x}-1}-\frac{1}{3}< 0\Rightarrow\frac{2\left(\sqrt{x}-4\right)}{3\left(\sqrt{x}-1\right)}< 0\Rightarrow1< x< 16\)
Bạn Kim Tuyến làm sai rùi , mk sửa lại :
a) 4x2 - 4x + 1 > 9
⇔ 4x2 - 4x - 8 > 0
⇔4x2 + 4x - 8x - 8 > 0
⇔ 4x( x + 1) -8( x + 1) > 0
⇔ ( x + 1)( 4x - 8) > 0
⇔ ( x + 1)( x - 2) > 0
Lập bảng xét dấu , ta có :
x x+1 x-2 -1 2 0 0 - + + - - + Tích số + - + 0 0 Vậy, nghiệm của BPT : x < -1 hoặc : x > 2
b) ( x - 5)( 7 - 2x ) < 0
Lập bảng xét dấu :
x x-5 7-2x tích số 7/2 5 0 0 0 0 - - + + - - - + - Vậy , nghiệm của BPT : x < 7/2 hoặc x > 5
a)
\(4x-10< 0\\ 4x< 10\\ x< \dfrac{10}{4}=\dfrac{5}{2}\)
b)
\(2x+x+12\ge0\\ 3x\ge-12\\ x\ge-\dfrac{12}{3}=-4\)
c)
\(x-5\ge3-x\\ 2x\ge8\\ x\ge4\)
d)
\(7-3x>9-x\\ -2>2x\\ x< -1\)
đ)
\(2x-\left(3-5x\right)\le4\left(x+3\right)\\ 2x-3+5x\le4x+12\\ 3x\le15\\ x\le5\)
e)
\(3x-6+x< 9-x\\ 5x< 15\\ x< 3\)
f)
\(2t-3+5t\ge4t+12\\ 3t\ge15\\ t\ge5\)
g)
\(3y-2\le2y-3\\ y\le-1\)
h)
\(3-4x+24+6x\ge x+27+3x\\ 0\ge2x\\ 0\ge x\)
i)
\(5-\left(6-x\right)\le4\left(3-2x\right)\\ 5-6+x\le12-8x\\ \\ 9x\le13\\ x\le\dfrac{13}{9}\)
k)
\(5\left(2x-3\right)-4\left(5x-7\right)\ge19-2\left(x+11\right)\\ 10x-15-20x+28\ge19-2x-22\\ 13-10x\ge-2x-3\\ -8x\ge-16\\ x\le\dfrac{-16}{-8}=2\)
l)
\(\dfrac{2x-5}{3}-\dfrac{3x-1}{2}< \dfrac{3-x}{5}-\dfrac{2x-1}{4}\\ \dfrac{40x-100}{60}-\dfrac{90x-30}{2}< \dfrac{36-12x}{60}-\dfrac{30x-15}{60}\\ \Rightarrow40x-100-90x+30< 36-12x-30x+15\\ 130-50x< 51-42x\\ 92x< -79\\ x< -\dfrac{79}{92}\)
m)
\(5x-\dfrac{3-2x}{2}>\dfrac{7x-5}{2}+x\\ \dfrac{10x}{2}-\dfrac{3-2x}{2}>\dfrac{7x-5}{2}+\dfrac{2x}{2}\\ \Rightarrow10x-3+2x>7x-5+2x\\ 12x-3>9x-5\\ 3x>-2\\ x>-\dfrac{2}{3}\)
n)
\(\dfrac{7x-2}{3}-2x< 5-\dfrac{x-2}{4}\\ \dfrac{28x-8}{12}-\dfrac{24x}{12}< \dfrac{60}{12}-\dfrac{3x-6}{12}\\ \Rightarrow28x-8-24x< 60-3x+6\\ 4x-8< -3x+66\\ 7x< 74\\ x< \dfrac{74}{7}\)
a) \(4x-10< 0\)
\(\Leftrightarrow4x< 10\)
\(\Leftrightarrow x< \dfrac{5}{2}\)
b) ???
c) \(x-5\ge3-x\)
\(\Leftrightarrow2x-5\ge3\)
\(\Leftrightarrow2x\ge8\)
\(\Leftrightarrow x\ge4\)
d) \(7-3x>9-x\)
\(\Leftrightarrow7-2x>9\)
\(\Leftrightarrow-2x>2\)
\(\Leftrightarrow x< -1\)
đ) ???
e) \(3x-6+x< 9-x\)
\(\Leftrightarrow4x-6< 9-x\)
\(\Leftrightarrow5x-6< 9\)
\(\Leftrightarrow5x< 15\)
\(\Leftrightarrow x< 3\)
f) ???
g) ???
h) \(3-4x+24+6x\ge x+27+3x\)
\(\Leftrightarrow2x+27\ge4x+27\)
\(\Leftrightarrow-2x\ge0\)
\(\Leftrightarrow x\le0\)
i) \(5-\left(6-x\right)\le4\left(3-2x\right)\)
\(\Leftrightarrow5-6+x\le12-8x\)
\(\Leftrightarrow x-1\le12-8x\)
\(\Leftrightarrow9x-1\le12\)
\(\Leftrightarrow9x\le13\)
\(\Leftrightarrow x\le\dfrac{13}{9}\)
k) \(5\left(2x-3\right)-4\left(5x-7\right)\ge19-2\left(x+11\right)\)
\(\Leftrightarrow10x-15-20x+28\ge19-2x-22\)
\(\Leftrightarrow-10x+23\ge-3-2x\)
\(\Leftrightarrow-8x+13\ge-3\)
\(\Leftrightarrow-8x\ge-16\)
\(\Leftrightarrow x\ge2\)
l) \(\dfrac{2x-5}{3}-\dfrac{3x-1}{2}< \dfrac{3-x}{5}-\dfrac{2x-1}{4}\)
\(\Leftrightarrow-\dfrac{5}{6}x-\dfrac{7}{6}< -\dfrac{7}{10}x+\dfrac{17}{20}\)
\(\Leftrightarrow-\dfrac{2}{15}x-\dfrac{7}{6}< \dfrac{17}{20}\)
\(\Leftrightarrow-\dfrac{2}{15}x< \dfrac{121}{60}\)
\(\Leftrightarrow x>-\dfrac{121}{8}\)
m, n) làm tương tự:
đáp án: m. \(x>-\dfrac{2}{3}\); n. \(x< \dfrac{74}{7}\)
a) \(B=\frac{x}{x+1}+\frac{2x-3}{x-1}-\frac{2x^2-x-3}{x^2-1}\)
\(B=\frac{x\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}+\frac{\left(2x-3\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}-\frac{2x^2-x-3}{\left(x-1\right)\left(x+1\right)}\)
\(B=\frac{\left(x^2-x\right)+\left(2x^2+2x-3x-3\right)-\left(2x^2-x-3\right)}{\left(x+1\right)\left(x-1\right)}\)
\(B=\frac{x^2-x+2x^2-x-3-2x^2+x+3}{\left(x+1\right)\left(x-1\right)}\)
\(B=\frac{x^2-x}{\left(x+1\right)\left(x-1\right)}\)
\(B=\frac{x\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}\)
\(B=\frac{x}{x+1}\)
MÌnh nghĩ đề câu b là với x>-4 mới đúng chứ
\(B=\frac{x}{x+1}+\frac{2x-3}{x-1}-\frac{2x^2-x-3}{\left(x^2-1\right)}.\)
\(=\frac{x\left(x-1\right)+\left(2x-3\right)\left(x+1\right)-2x^2+x+3}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x^2-x+2x^2-x-3-2x^2+x+3}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x^2-x}{\left(x-1\right)\left(x+1\right)}=\frac{x\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}=\frac{x}{x+1}\)
\(\Rightarrow A.B=\frac{x}{\left(x+1\right)}.\frac{x\left(x+1\right)}{\left(x-2\right)}=\frac{x^2}{\left(x-2\right)}=\frac{x^2-4+4}{\left(x-2\right)}\)
\(=\frac{\left(x-2\right)\left(x+2\right)+4}{\left(x-2\right)}=x+2+\frac{4}{x-2}=x-2+\frac{4}{x-2}+4\)
Áp dụng BĐT Cô - Si cho 2 số dương \(x-2;\frac{4}{x-2}\)ta có :
\(x-2+\frac{4}{x-2}\ge2\sqrt{\frac{\left(x-2\right).4}{x-2}}=2\sqrt{4}=4\)
\(\Rightarrow x-2+\frac{4}{x-2}\ge4\Rightarrow x-2+\frac{4}{x-2}+4\ge8\)
Hay \(S_{min}=4\Leftrightarrow x-2=\frac{4}{x-2}\)
\(\Rightarrow\frac{\left(x-2\right)^2}{\left(x-2\right)}=\frac{4}{x-2}\Rightarrow x^2+4x+4=4\)
\(\Rightarrow x^2+4x=0\Rightarrow x\left(x+4\right)=0\)
\(\Rightarrow\orbr{\begin{cases}x=0\left(tm\right)\\x=-4\left(ktm\right)\end{cases}}\)\(\Rightarrow...\)
bài này đề bài là chứng minh hay là giải bất phương trình vậy bạn
a) \(x^2-4x+5\)
= \(\left(x^2-2.2x+4\right)+1\)
= \(\left(x-2\right)^2+1\)
Ta co: \(\left(x-2\right)^2>=0\)
=>\(\left(x-2\right)^2+1>=1>0\)
b) \(x^2-4xy+5y^2\)
=\(\left(x^2-4xy+4y^2\right)+y^2\)
= \(\left(x-2y\right)^2+y^2\)
Ta co: \(\left(x-2y\right)^2>=0\)
\(y^2>=0\)
=> \(\left(x-2y\right)^2+y^2>=0\)
c) \(3-2x-x^2\)
= \(-\left(x^2+2x\right)+3\)
= \(-\left(x^2+2.1x+1-1\right)+3\)
= \(-\left(x+1\right)^2+4\)
=
Hình như câu này sai đề ...
a)\(x^2+4y^2-2x+4y+2\)
\(=\left(x^2-2x+1\right)+\left(4y^2+4y+1\right)\)
\(=\left(x-1\right)^2+\left(2y+1\right)^2\ge0\)(đúng)
b) Sửa đề
\(3y^2+x^2+2xy+2x+6y+3\)
\(=\left(x^2+y^2+2xy\right)+2y^2+2x+6y+3\)
\(=\left(x+y\right)^2+2\left(x+y\right)+1+2y^2+4y+2\)
\(=\left(x+y+1\right)^2+2\left(y+1\right)^2\ge0\) (đúng)