1.

- Với \(x\ge\frac{1}{2}\Rightarrow2x-1\le x+2\Rightarrow x\le3\Rightarrow\frac{1}{2}\le x\le3\)

- Với \(x< \frac{1}{2}\Rightarrow1-2x\le x+2\Rightarrow3x\ge-1\Rightarrow x\ge-\frac{1}{3}\)

Vậy nghiệm của BPT là \(-\frac{1}{3}\le x\le3\)

2.

Để pt có 2 nghiệm trái dấu

\(\Leftrightarrow ac< 0\Leftrightarrow\left(m+2\right)\left(2m-3\right)< 0\Rightarrow-2< m< \frac{3}{2}\)

3.

\(5x-1>\frac{2x}{5}+3\Leftrightarrow5x-\frac{2x}{5}>4\Leftrightarrow\frac{23}{5}x>4\Rightarrow x>\frac{20}{23}\)

4.

\(4x^2+4x+1-3x+9>4x^2+10\)

\(\Leftrightarrow x>0\)

5.

\(1< \frac{1}{1-x}\Leftrightarrow\frac{1}{1-x}-1>0\Leftrightarrow\frac{x}{1-x}>0\Rightarrow0< x< 1\)

6.

\(\frac{\left(x-5\right)^2\left(x-3\right)}{x+1}\le0\Rightarrow\left[{}\begin{matrix}x=5\\-1< x\le3\end{matrix}\right.\)

\(0\le x\le\dfrac{5}{2}\)

Tập nghiệm của BPT là \(\left[{}\begin{matrix}x\le0\\1< x\le\dfrac{5}{2}\end{matrix}\right.\)

Câu 1: 

Ta có: \(\Delta=\left[-2\left(m+2\right)\right]^2-4\cdot m\cdot\left(2+3m\right)\)

\(\Leftrightarrow\Delta=\left(2m+4\right)^2-4m\left(2+3m\right)\)

\(\Leftrightarrow\Delta=4m^2+16m+16-8m-12m^2\)

\(\Leftrightarrow\Delta=-8m^2+8m+16\)

\(\Leftrightarrow\Delta=-8\left(m^2-m-2\right)\)

Để phương trình vô nghiệm thì \(\Delta< 0\)

\(\Leftrightarrow m^2-m-2>0\)

\(\Leftrightarrow\left(m-2\right)\left(m+1\right)>0\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}m-2>0\\m+1>0\end{matrix}\right.\\\left\{{}\begin{matrix}m-2< 0\\m+1< 0\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}m>2\\m>-1\end{matrix}\right.\\\left\{{}\begin{matrix}m< 2\\m< -1\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m>2\\m< -1\end{matrix}\right.\)

Câu 1 

Để pt vô nghiệm \(\Rightarrow\Delta'=\left(m+2\right)^2-\left(3m+2\right)m=m^2+4m+4-3m^2-2m=-2m^2+2m+4=-2\left(m^2-m-2\right)=-2\left(m+1\right)\left(m-2\right)< 0\) \(\Leftrightarrow\left(m+1\right)\left(m-2\right)>0\Leftrightarrow\left[{}\begin{matrix}m< -1\\m>2\end{matrix}\right.\)

x^2 + 10 ≤ 2x^2+1/x^2 -8

<=> x^2 + 10   -    2x^2+1/x^2 -8 ≤ 0

<=> (x^2+10)(x^2-8)-2x^2+1/x^2-8 ≤ 0

<=> x^4+2x^2-80-2x^2+1/x^2-8 ≤ 0

<=> x^4-81/x^2-8 ≤ 0

<=> (x^2+9)(x^2-9)/x^2-8 ≤ 0

<=> x^2-9/x^2-8 (do x^2 + 9 >0)

<=> x^2-9≤0, x^2-8>0

<=> -3≤x≤3, x<-2√2 hoặc x>2√2

<=> -3≤x<-2√2 hoặc 2√2<x≤3

=> bpt có 2 nghiệm nguyên là -3, 3