\(C=-2\left|\dfrac{1}{3}x+4\right|+1\dfrac{2}{3}\)

\(\Rightarrow C=-2\left|\dfrac{1}{3}x+4\right|+\dfrac{5}{3}\)

mà \(-2\left|\dfrac{1}{3}x+4\right|\le0,\forall x\)

\(\Rightarrow C=-2\left|\dfrac{1}{3}x+4\right|+\dfrac{5}{3}\le\dfrac{5}{3}\)

\(\Rightarrow GTLN\left(C\right)=\dfrac{5}{3}\left(tạix=-12\right)\)

Ta có :

\(\widehat{xOy}+\widehat{yOz}=180^o\) (2 góc kề bù)

\(\Rightarrow\widehat{yOz}=180^o-\widehat{xOy}\)

\(\Rightarrow\widehat{yOz}=180^o-80^o\)

\(\Rightarrow\widehat{yOz}=100^o\)

Ta lại có :

\(\widehat{tOz}=\widehat{tOy}+\widehat{yOz}\)

mà \(\widehat{tOy}=\dfrac{1}{2}\widehat{xOy}=\dfrac{1}{2}.80^o=40^o\) (Ot là phân giác góc xOy)

\(\Rightarrow\widehat{tOz}=40^o+100^o\)

\(\Rightarrow\widehat{tOz}=140^o\)

\(\widehat{xOt}=\dfrac{1}{2}\widehat{xOy}=\dfrac{1}{2}.80^o=40^o\) (Ot là phân giác góc xOy)

\(x=7\Rightarrow8=x+1\left(1\right)\)

Thay \(1\) vào \(F\) ta có:

\(F=x^{2006}-\left(x+1\right)^{2005}+\left(x+1\right)^{2004}-...+\left(x+1\right)x^2-\left(x+1\right)x-5\)

\(F=x^{2006}-x^{2006}-x^{2005}+x^{2005}+x^{2004}-...+x^3+x^2-x^2-x-5\)

\(F=-7-5\)

\(\Rightarrow F=-12\)

\(\left(1-\dfrac{1}{1+2}\right)\cdot\left(1-\dfrac{1}{1+2+3}\right)\cdot\left(\dfrac{1}{1+2+3+...+2006}\right)\)

\(=\left(1-\dfrac{1}{3}\right)\cdot\left(1-\dfrac{1}{6}\right)\cdot\left\{\dfrac{1}{\left(2006+1\right)\left[\left(2006-1\right):1+1\right]}\right\}\)

\(=\dfrac{2}{3}\cdot\dfrac{5}{6}\cdot\dfrac{1}{2007\cdot2006}\)

\(=\dfrac{10}{18}\cdot\dfrac{1}{4026042}\)

\(=\dfrac{5}{9}\cdot\dfrac{1}{4026042}\)

\(=\dfrac{5}{36234378}\)

a) \(x+\left|x-2\right|=7\)

\(\Leftrightarrow\left|x-2\right|=7-x\)

\(\Leftrightarrow\left[{}\begin{matrix}x-2=7-x\\x-2=-7+x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=9\\0x=-5\left(loại\right)\end{matrix}\right.\) \(\Leftrightarrow x=\dfrac{9}{2}\)

b) \(\left|x-3\right|+\left|x-5\right|=9\left(1\right)\)

Ta thấy :

\(\left|x-3\right|+\left|x-5\right|\ge\left|x-3+x-5\right|=\left|2x-8\right|\)

\(pt\left(1\right)\Leftrightarrow\left|2x-8\right|=9\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-8=9\\2x-8=-9\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=17\\2x=-1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{17}{2}\\x=-\dfrac{1}{2}\end{matrix}\right.\)

c) \(\left|x-1\right|+\left|x+1\right|=10\left(1\right)\)

Ta thấy :

\(\left|x-1\right|+\left|x+1\right|\ge\left|x-1+x+1\right|=\left|2x\right|\)

\(pt\left(1\right)\Leftrightarrow\left|2x\right|=10\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=10\\2x=-10\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=5\\x=-5\end{matrix}\right.\)

a) \(x+\left|x-2\right|=7\)

\(\Rightarrow\left\{{}\begin{matrix}x+\left(x-2\right)=7\left(x\ge2\right)\\x-\left(x-2\right)=7\left(x< 2\right)\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x+x-2=7\\x-x+2=7\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}2x-2=7\\2=7\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}2x=9\\x\in\varnothing\end{matrix}\right.\)

\(\Rightarrow2x=-9\)

\(\Rightarrow x=-\dfrac{9}{2}\)

\(5^{x-3}=25^9\)

\(\Rightarrow5^{x-3}=\left(5^2\right)^9\)

\(\Rightarrow5^{x-3}=5^{18}\)

\(\Rightarrow x-3=18\)

\(\Rightarrow x=18+3\)

\(\Rightarrow x=21\)

Vậy: x=21

\(5x^2=1620\)

\(\Rightarrow x^2=1620:5\)

\(\Rightarrow x^2=324\)

\(\Rightarrow x^2=18^2\)

\(\Rightarrow\left[{}\begin{matrix}x=18\\x=-18\end{matrix}\right.\)

Vậy: x=18 hoặc x=-18

\(5^x+10x=625+10x\)

\(\Rightarrow5^x=625+10x-10x\)

\(\Rightarrow5^x=625\)

\(\Rightarrow5^x=5^4\)

\(\Rightarrow x=4\)

Vậy x=4

\(2^x:8=4\)

\(\Rightarrow2^x=4\cdot8\)

\(\Rightarrow2^x=32\)

\(\Rightarrow2^x=2^5\)

\(x=5\)

Vậy: x=5

\(2^x:8=4\)

\(\Rightarrow2^x=4.8\)

\(\Rightarrow2^x=32\)

\(\Rightarrow2^5=32\)

\(\Rightarrow x=5\)

A = 1 + 3 + 5 + ... + 299

a) Số số hạng của A:

(299 - 1) : 2 + 1 = 150 (số)

A = (299 + 1) . 150 : 2 = 22500

b) Số hạng thứ 74 của A:

1 + 73 × 2 = 147