\(a,\)\(A=\left\{x\in R|x< 3\right\}\Rightarrow A=\left(\text{ -∞;3}\right)\)

\(B=\left\{-1;0;1;2;3;4;5\right\}\)

\(\Rightarrow A\cap B=\left\{-1;0;1;2\right\}\)

\(b,x=-1\Rightarrow y=1-2\left(-1\right)+m=m+3\) 

\(x=1\Rightarrow y=1-2+m=m-1\)

\(\Rightarrow C=(m-1;m+3]\subset A\)

\(\Rightarrow C\subset A\Leftrightarrow m+3< 3\Leftrightarrow m< 0\)

(Bấm máy tính tìm nghiệm)

\(A=\left\{-2;-1;2\right\}\)

\(B=\left\{0;1;2;3\right\}\)

\(x^2+y^2-2x-4y-4=0\\ \Leftrightarrow\left(x-1\right)^2+\left(y-2\right)^2-9=0\\ \Leftrightarrow\left(x-1\right)^2+\left(y-2\right)^2=9=0^2+3^2=0^2+\left(-3\right)^2\\ \Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-1=0\\y-2=3\end{matrix}\right.\\\left\{{}\begin{matrix}x-1=3\\y-2=0\end{matrix}\right.\\\left\{{}\begin{matrix}x-1=0\\y-2=-3\end{matrix}\right.\\\left\{{}\begin{matrix}x-1=-3\\y-2=0\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=1\\y=5\end{matrix}\right.\\\left\{{}\begin{matrix}x=4\\y=2\end{matrix}\right.\\\left\{{}\begin{matrix}x=1\\y=-1\end{matrix}\right.\\\left\{{}\begin{matrix}x=-2\\y=2\end{matrix}\right.\end{matrix}\right.\\ \Leftrightarrow-2\le x\le4\left(y\in R\right)\)

Ta có \(S=3x+4y\)

Mà \(x\ge-2;y\ge-1\Leftrightarrow S\ge3\cdot\left(-2\right)+4\cdot\left(-1\right)=-6-4=-10\)

Vậy GTNN của S là \(-10\Leftrightarrow\left\{{}\begin{matrix}x=-2\\y=-1\end{matrix}\right.\)

Lời giải:

ĐKĐB $\Leftrightarrow (x^2-2x+1)+(y^2-4y+4)-9=0$

$\Leftrightarrow (x-1)^2+(y-2)^2-9=0$

$\Rightarrow (x-1)^2=9-(y-2)^2\leq 9$

$\Rightarrow -3\leq x-1\leq 3$

$\Leftrightarrow -2\leq x\leq 4$

-------------

Đặt $x-1=a; y-2=b$ thì bài toán trở thành:
Cho $a,b$ thực thỏa mãn $a^2+b^2=9$

Tìm min $S=3a+4b+11$

Áp dụng BĐT Bunhiacopxky:

$(3a+4b)^2\leq (a^2+b^2)(3^2+4^2)=9.25$

$\Rightarrow -15\leq 3a+4b\leq 15$

$\Rightarrow 3a+4b\geq -15$

$\Rightarrow S=3a+4b+11\geq -4$

Vậy $S_{\min}=-4$ khi $x=\frac{-4}{5}; y=\frac{-1}{5}$

A=[-4;4]

B=[-3;2)

\(A\cap B\)=[-3;2)

A\B=[-4;-3)

B\A=\(\varnothing\)

\(E=\left\{-5;-4;-3;-2;-1;0;1;2;3;4;5\right\}\)

\(A=\left\{1;-4\right\}\)

\(B=\left\{2;-1\right\}\)

a) Với mọi x thuộc A đều thuộc E \(\Rightarrow A\subset E\)

Với mọi x thuộc B đều thuộc E \(\Rightarrow B\subset E\)

b) \(A\cap B=\varnothing\)

\(\Rightarrow E\backslash\left(A\cap B\right)=\left\{-5;-4;-3;-2;-1;0;1;2;3;4;5\right\}\)

\(A\cup B=\left\{-4;-1;1;2\right\}\)

\(\Rightarrow E\backslash\left(A\cup B\right)=\left\{-5;-3;-2;0;3;4;5\right\}\)

\(\Rightarrow E\backslash\left(A\cup B\right)\subset E\backslash\left(A\cap B\right)\)

A=(6-2x)(12-3y)(2x+3y)/6

<=(6-2x+12-3y+2x+3y)³/(6.27)

=18³/(6.27)=36

1.a.

\(\left(x+1\right)\left(x+2\right)\left(x-2\right)\left(x+5\right)\ge m\)

\(\Leftrightarrow\left(x^2+3x+2\right)\left(x^2+3x-10\right)\ge m\)

Đặt \(x^2+3x-10=t\ge-\dfrac{49}{4}\)

\(\Rightarrow\left(t+2\right)t\ge m\Leftrightarrow t^2+2t\ge m\)

Xét \(f\left(t\right)=t^2+2t\) với \(t\ge-\dfrac{49}{4}\)

\(-\dfrac{b}{2a}=-1\) ; \(f\left(-1\right)=-1\) ; \(f\left(-\dfrac{49}{4}\right)=\dfrac{2009}{16}\)

\(\Rightarrow f\left(t\right)\ge-1\)

\(\Rightarrow\) BPT đúng với mọi x khi \(m\le-1\)

Có 30 giá trị nguyên của m

1b.

Với \(x=0\)  BPT luôn đúng

Với \(x\ne0\) BPT tương đương:

\(\dfrac{\left(x^2-2x+4\right)\left(x^2+3x+4\right)}{x^2}\ge m\)

\(\Leftrightarrow\left(x+\dfrac{4}{x}-2\right)\left(x+\dfrac{4}{x}+3\right)\ge m\)

Đặt \(x+\dfrac{4}{x}-2=t\) \(\Rightarrow\left[{}\begin{matrix}t\ge2\\t\le-6\end{matrix}\right.\)

\(\Rightarrow t\left(t+5\right)\ge m\Leftrightarrow t^2+5t\ge m\)

Xét hàm \(f\left(t\right)=t^2+5t\) trên \(D=(-\infty;-6]\cup[2;+\infty)\)

\(-\dfrac{b}{2a}=-\dfrac{5}{2}\notin D\) ; \(f\left(-6\right)=6\) ; \(f\left(2\right)=14\)

\(\Rightarrow f\left(t\right)\ge6\)

\(\Rightarrow m\le6\)

Vậy có 37 giá trị nguyên của m thỏa mãn

2.

Xét với \(x\ge1\)

\(m\left(x+1\right)+3\left(x-1\right)-2\sqrt{x^2-1}=0\)

\(\Leftrightarrow m+3\left(\dfrac{x-1}{x+1}\right)-2\sqrt{\dfrac{x-1}{x+1}}=0\)

Đặt \(\sqrt{\dfrac{x-1}{x+1}}=t\Rightarrow0\le t< 1\)

\(\Rightarrow m+3t^2-2t=0\)

\(\Leftrightarrow3t^2-2t=-m\)

Xét hàm \(f\left(t\right)=3t^2-2t\) trên \(D=[0;1)\)

\(-\dfrac{b}{2a}=\dfrac{1}{3}\in D\) ; \(f\left(0\right)=0\) ; \(f\left(\dfrac{1}{3}\right)=-\dfrac{1}{3}\) ; \(f\left(1\right)=1\)

\(\Rightarrow-\dfrac{1}{3}\le f\left(t\right)< 1\)

\(\Rightarrow\) Pt có nghiệm khi \(-\dfrac{1}{3}\le-m< 1\)

\(\Leftrightarrow-1< m\le\dfrac{1}{3}\)

a, \(A\cup B=(-4;5]\)

\(A\cap B=[-3;4)\)

\(A\backslash B=\left[4;5\right]\)

\(B\backslash A=\left(-4;-3\right)\)

b, \(A\cup B=\left(-3;7\right)\)

\(A\cap B=[1;2)\cup(3;5]\)

\(A\backslash B=\left[2;3\right]\)

\(B\backslash A=\left(-3;1\right)\cup\left(5;7\right)\)

c, \(A\cup B=\left[\dfrac{1}{2};3\right]\)

\(A\cap B=\left[1;\dfrac{3}{2}\right]\)

\(A\backslash B=[\dfrac{1}{2};1)\)

\(B\backslash A=(\dfrac{3}{2};3]\)

d, \(A\cup B=(-5;2]\cup(3;6]\)

\(A\cap B=\left\{0\right\}\cup[4;5)\)

\(A\backslash B=(0;2]\cup\left[-5;6\right]\)

\(B\backslash A=[-5;0)\cup\left(3;4\right)\)

a) Khoảng \(\left( { - 2;3} \right)\)

b) Đoạn \(\left[ {1;10} \right]\)

c) Nửa khoảng \(\left( {\left. { - 5;\sqrt 3 } \right]} \right.\)

d) Nửa khoảng \(\left. {\left[ {\pi ;4} \right.} \right)\)

e) Khoảng \(\left( { - \infty ;\frac{1}{4}} \right)\)

g) Nửa khoảng \(\left[ {\left. {\frac{\pi }{2}; + \infty } \right)} \right.\)