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\(a,\dfrac{x-3}{x}=\dfrac{x-3}{x+3}\)\(\left(đk:x\ne0,-3\right)\)
\(\Leftrightarrow\dfrac{x-3}{x}-\dfrac{x-3}{x+3}=0\)
\(\Leftrightarrow\dfrac{\left(x-3\right)\left(x+3\right)-x\left(x-3\right)}{x\left(x+3\right)}=0\)
\(\Leftrightarrow x^2-9-x^2+3x=0\)
\(\Leftrightarrow3x-9=0\)
\(\Leftrightarrow3x=9\)
\(\Leftrightarrow x=3\left(n\right)\)
Vậy \(S=\left\{3\right\}\)
\(b,\dfrac{4x-3}{4}>\dfrac{3x-5}{3}-\dfrac{2x-7}{12}\)
\(\Leftrightarrow\dfrac{4x-3}{4}-\dfrac{3x-5}{3}+\dfrac{2x-7}{12}>0\)
\(\Leftrightarrow\dfrac{3\left(4x-3\right)-4\left(3x-5\right)+2x-7}{12}>0\)
\(\Leftrightarrow12x-9-12x+20+2x-7>0\)
\(\Leftrightarrow2x+4>0\)
\(\Leftrightarrow2x>-4\)
\(\Leftrightarrow x>-2\)
a.
\(3\sqrt{-x^2+x+6}\ge2\left(1-2x\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}-x^2+x+6\ge0\\1-2x< 0\end{matrix}\right.\\\left\{{}\begin{matrix}1-2x\ge0\\9\left(-x^2+x+6\right)\ge4\left(1-2x\right)^2\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}-2\le x\le3\\x>\dfrac{1}{2}\end{matrix}\right.\\\left\{{}\begin{matrix}x\le\dfrac{1}{2}\\25\left(x^2-x-2\right)\le0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{1}{2}< x\le3\\\left\{{}\begin{matrix}x\le\dfrac{1}{2}\\-1\le x\le2\end{matrix}\right.\end{matrix}\right.\)
\(\Rightarrow-1\le x\le3\)
b.
ĐKXĐ: \(x\ge0\)
\(\Leftrightarrow\sqrt{2x^2+8x+5}-4\sqrt{x}+\sqrt{2x^2-4x+5}-2\sqrt{x}=0\)
\(\Leftrightarrow\dfrac{2x^2+8x+5-16x}{\sqrt{2x^2+8x+5}+4\sqrt{x}}+\dfrac{2x^2-4x+5-4x}{\sqrt{2x^2-4x+5}+2\sqrt{x}}=0\)
\(\Leftrightarrow\dfrac{2x^2-8x+5}{\sqrt{2x^2+8x+5}+4\sqrt{x}}+\dfrac{2x^2-8x+5}{\sqrt{2x^2-4x+5}+2\sqrt{x}}=0\)
\(\Leftrightarrow\left(2x^2-8x+5\right)\left(\dfrac{1}{\sqrt{2x^2+8x+5}+4\sqrt{x}}+\dfrac{1}{\sqrt{2x^2-4x+5}+2\sqrt{x}}\right)=0\)
\(\Leftrightarrow2x^2-8x+5=0\)
\(\Leftrightarrow x=\dfrac{4\pm\sqrt{6}}{2}\)
a)
\(2x-1+5\left(3-x\right)>0\\ 2x-2+15-5x>0\\ -3x+13>0\\ x< \dfrac{13}{3}.\)
\(\dfrac{3}{x-2}\ge\dfrac{5}{2x-1}.\\ \Leftrightarrow\dfrac{3}{x-2}-\dfrac{5}{2x-1}\ge0.\\ \Leftrightarrow\dfrac{6x-3-5x+10}{\left(x-2\right)\left(2x-1\right)}\ge0.\\ \Leftrightarrow\dfrac{x+7}{\left(x-2\right)\left(2x-1\right)}\ge0.\)
Ta có:
\(x+7=0.\Leftrightarrow x=-7.\\ x-2=0.\Leftrightarrow x=2.\\ 2x-1=0.\Leftrightarrow x=\dfrac{1}{2}.\)
Đặt \(f\left(x\right)=\dfrac{x+7}{\left(x-2\right)\left(2x-1\right)}.\)
Bảng xét dấu:
\(x\) \(-\infty\) \(-7\) \(\dfrac{1}{2}\) \(2\) \(+\infty\)
\(x+7\) - 0 + | + | +
\(x-2\) - | - | - 0 +
\(2x-1\) - | - 0 + | +
\(f\left(x\right)\) - 0 + || - || +
Vậy \(f\left(x\right)\ge0.\Leftrightarrow x\in[-7;\dfrac{1}{2})\cup\left(2;+\infty\right).\)
a: =>4x^2-24x+36-4x^2+4x-1<10
=>-20x<10-35=-25
=>x>=5/4
b: =>x(x^2-25)-x^3-8<=3
=>x^3-25x-x^3-8<=3
=>-25x<=11
=>x>=-11/25
a: Ta có: \(3x-5\ge2\left(x-6\right)-12\)
\(\Leftrightarrow3x-5\ge2x-24\)
hay \(x\ge-19\)
b: Ta có: \(2\left(5-2x\right)\ge3-x\)
\(\Leftrightarrow10-4x-3+x\ge0\)
\(\Leftrightarrow-3x\ge-7\)
hay \(x\le\dfrac{7}{3}\)
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\(\dfrac{3}{x-2}\ge\dfrac{5}{2x-1}\)
ĐKXĐ: x ≠ 2; \(x\ne\dfrac{1}{2}\)
\(\dfrac{3}{x-2}\ge\dfrac{5}{2x-1}\)
\(\Leftrightarrow\dfrac{3}{x-2}-\dfrac{5}{2x-1}\ge0\)
\(\Leftrightarrow\dfrac{3\left(2x-1\right)-5\left(x-2\right)}{\left(x-2\right)\left(2x-1\right)}\ge0\)
\(\Leftrightarrow\dfrac{x+7}{\left(x-2\right)\left(2x-1\right)}\ge0\)
*Với: \(\dfrac{x+7}{\left(x-2\right)\left(2x-1\right)}=0\)
=> x + 7 = 0
<=> x =-7
*Với \(\dfrac{x+7}{\left(x-2\right)\left(2x-1\right)}>0\) (1)
Ta lâpj bảng xét dấu:
x
-7
1/2
2
X + 7
-
0
+
|
+
|
+
2x – 1
-
|
-
0
+
|
+
X - 2
-
|
-
|
-
0
+
BĐT (1)
-
0
+
||
-
||
+
Từ bảng trên ta có thể thấy: \(\dfrac{x+7}{\left(x-2\right)\left(2x-1\right)}>0\) khi -7 < x < 1/2 hoăcj x > 2
Vayj:.............