Chương 4: BẤT ĐẲNG THỨC, BẤT PHƯƠNG TRÌNH

Đề bài sai, phản ví dụ: \(x=y=\dfrac{1}{16};z=256\)

Nói chung, chỉ cần 2 biến đủ nhỏ là BĐT này đều sai

1.

\(\Leftrightarrow\left\{{}\begin{matrix}2x+1\ge0\\\left(2x+1\right)^2\ge x^2+3x+5\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-\dfrac{1}{2}\\4x^2+4x+1\ge x^2+3x+5\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-\dfrac{1}{2}\\3x^2+x-4\ge0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-\dfrac{1}{2}\\\left[{}\begin{matrix}x\le-\dfrac{4}{3}\\x\ge1\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow x\ge1\)

2.

\(\Leftrightarrow\left\{{}\begin{matrix}x-3>0\\x^2+x-12\ge0\\\left(x-3\right)^2>x^2+x-12\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x>3\\x^2+x-12\ge0\\7x< 21\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x>3\\\left[{}\begin{matrix}x\le-4\\x\ge3\end{matrix}\right.\\x< 3\end{matrix}\right.\)

\(\Rightarrow x\in\varnothing\)

BPT đã cho vô nghiệm

3.

\(\Leftrightarrow x-1>\sqrt{2x^2-3x-5}\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-1>0\\2x^2-3x-5\ge0\\\left(x-1\right)^2>2x^2-3x-5\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x>1\\2x^2-3x-5\ge0\\x^2-x-6< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x>1\\\left[{}\begin{matrix}x\le-1\\x\ge\dfrac{5}{3}\end{matrix}\right.\\-2< x< 3\end{matrix}\right.\)

\(\Leftrightarrow\dfrac{5}{3}\le x< 3\)

\(ĐKXĐ:x\ge-\dfrac{3}{2}\)

Bất phương trình tương đương :

\(2x+3+x+2+2\sqrt{\left(2x+3\right)\left(x+2\right)}\le1\)

\(\Leftrightarrow2\sqrt{\left(2x+3\right)\left(x+2\right)}\le-3x-4\)

\(\Leftrightarrow4.\left(2x+3\right)\left(x+2\right)\le\left(-3x-4\right)^2\)

\(\Leftrightarrow4.\left(2x^2+7x+6\right)\le9x^2+16+24x\)

\(\Leftrightarrow x^2-4x-8\ge0\)

\(\Leftrightarrow\left[{}\begin{matrix}x\ge2+2\sqrt{3}\\x\le2-2\sqrt{3}\end{matrix}\right.\). Kết hợp với ĐKXĐ ....

P/s : E không chắc lắm .....

Bài 1 :

a,b,d tương tự

b, - Đặt \(f_{\left(x\right)}=\left(2x-7\right)\left(4-5x\right)\)

- \(f\left(x\right)=0\Rightarrow\left[{}\begin{matrix}x=\dfrac{7}{2}\\x=\dfrac{4}{5}\end{matrix}\right.\)

- Lập bảng xét dấu :

- Từ bảng xét dấu : Để \(f\left(x\right)\ge0\)

\(\Rightarrow x=\left[\dfrac{4}{5};\dfrac{7}{2}\right]\)

Vậy .....

c, e, f tương tự

e, Đặt \(f\left(x\right)=x^3+8x^2+17x+10\)  \(=x^3+5x^2+3x^2+15x+2x+10\)

\(=\left(x+5\right)\left(x^2+3x+2\right)\)

- Cho \(f\left(x\right)=0\Rightarrow\left[{}\begin{matrix}x=-1\\x=-2\\x=-5\end{matrix}\right.\)

- Lập bảng xét dấu :

- Từ bảng xét dấu : Để f(x) < 0

\(\Rightarrow x=\left(-\infty;-5\right)\cup\left(-2;-1\right)\)

- Gửi lẻ câu đại diện ra để tham khảo thôi bn ơi

\(\dfrac{1-2x}{3-\left|x-1\right|}-3\le0\Leftrightarrow\dfrac{3\left|x-1\right|-2x-8}{3-\left|x-1\right|}\le0\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge1\\\dfrac{3\left(x-1\right)-2x-8}{3-\left(x-1\right)}\le0\end{matrix}\right.\\\left\{{}\begin{matrix}x< 1\\\dfrac{3\left(1-x\right)-2x-8}{3-\left(1-x\right)}\le0\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge1\\\dfrac{x-11}{4-x}\le0\end{matrix}\right.\\\left\{{}\begin{matrix}x< 1\\\dfrac{x+1}{x+2}\ge0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge0\\\left[{}\begin{matrix}x< 4\\x\ge11\end{matrix}\right.\end{matrix}\right.\\\left\{{}\begin{matrix}x< 1\\\left[{}\begin{matrix}x< -2\\x\ge-1\end{matrix}\right.\end{matrix}\right.\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\left[{}\begin{matrix}0\le x< 4\\x\ge11\end{matrix}\right.\\\left[{}\begin{matrix}x< -2\\-1\le x< 1\end{matrix}\right.\end{matrix}\right.\) 

1.

\(\Leftrightarrow\Delta=\left(m-2\right)^2-8\left(-m+4\right)\le0\)

\(\Leftrightarrow m^2+4m-28\le0\)

\(\Rightarrow-2-4\sqrt{2}\le m\le-2+4\sqrt{2}\)

2.

\(\Delta'=\left(m+1\right)^2-3m^2-m< 0\)

\(\Leftrightarrow m^2+2m+1-3m^2-m< 0\)

\(\Leftrightarrow-2m^2-m+1< 0\)

\(\Rightarrow\left[{}\begin{matrix}m>\dfrac{1}{2}\\m< -1\end{matrix}\right.\)

3.

\(\Leftrightarrow x^2-\left(3m-2\right)x+2m^2-5m-2>0\) nghiệm đúng với mọi x

\(\Leftrightarrow\Delta=\left(3m-2\right)^2-4\left(2m^2-5m-2\right)< 0\)

\(\Leftrightarrow m^2+8m+12< 0\)

\(\Leftrightarrow-6< m< -2\)

4.\(\Leftrightarrow-x^2-4\left(m+1\right)x+m-5\le0\) nghiệm đúng với mọi x

\(\Leftrightarrow\Delta'=4\left(m+1\right)^2+m-5\le0\)

\(\Leftrightarrow4m^2+9m-1\le0\)

\(\Rightarrow\dfrac{-9-\sqrt{87}}{8}\le m\le\dfrac{-9+\sqrt{87}}{8}\)