Question

Giải phương trình

a) |x-3|+|5-x|=4

b) |x-1|+|x-2|+3|x-4|=11

c)|x-1|+|x-2|+|x-3|+|x-4|+30=5x

Answer 1

a,\(\left|x-3\right|+\left|5-x\right|=4\)

\(\Leftrightarrow\left[{}\begin{matrix}x-3+5-x=4\\-x+3+5-x=4\\x-3-5+x=4\\-x-3-5+x=4\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}2=4\left(ktm\right)\\-2x+8=4\\2x-8=4\\-8=4\left(ktm\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=6\end{matrix}\right.\)

Vậy...

P/s:Các câu còn lại tương tự nhé!!!

Answer 2

a) \(\left|x-3\right|+\left|5-x\right|=4\)

Lập bảng xét dấu:

+) Xét \(x< 3\Leftrightarrow\left(3-x\right)+\left(x-5\right)=4\)

\(\Leftrightarrow3-x+x-5=4\\ \Leftrightarrow-2=4\left(\text{Vô lý}\right)\)

+) Xét \(3\le x< 5\Leftrightarrow\left(x-3\right)+\left(x-5\right)=4\)

\(\Leftrightarrow x-3+x-5=4\\ \Leftrightarrow2x=12\\ \Leftrightarrow x=6\left(KTM\right)\)

+) Xét \(x\ge5\Leftrightarrow\left(x-3\right)+\left(5-x\right)=4\)

\(\Leftrightarrow x-3+5-x=4\\ \Leftrightarrow2=4\left(\text{Vô lý}\right)\)

Vậy phương trình vô nghiệm.

Answer 3

\(\text{b) }\left|x-1\right|+\left|x-2\right|+3\left|x-4\right|=11\)

Lập bảng xét dấu:

+) Xét \(x< 1\Leftrightarrow\left(1-x\right)+\left(2-x\right)+3\left(4-x\right)=11\)

\(\Leftrightarrow1-x+2-x+12-3x=11\\ \Leftrightarrow15-5x=11\\ \Leftrightarrow5x=4\\ \Leftrightarrow x=\dfrac{4}{5}\left(TM\right)\)

+) Xét \(1\le x< 2\Leftrightarrow\left(x-1\right)+\left(2-x\right)+3\left(4-x\right)=11\)

\(\Leftrightarrow x-1+2-x+12-3x=11\\ \Leftrightarrow13-3x=11\\ \Leftrightarrow3x=2\\ \Leftrightarrow x=\dfrac{2}{3}\left(KTM\right)\)

+) Xét \(2\le x< 4\Leftrightarrow\left(x-1\right)+\left(x-2\right)+3\left(4-x\right)=11\)

\(\Leftrightarrow x-1+x-2+12-3x=11\\ \Leftrightarrow9-x=11\\ \Leftrightarrow x=-2\left(KTM\right)\)

+) Xét \(x\ge4\Leftrightarrow\left(x-1\right)+\left(x-2\right)+3\left(x-4\right)=11\)

\(\Leftrightarrow x-1+x-2+3x-12=11\\ \Leftrightarrow5x-15=11\\ \Leftrightarrow5x=26\\ \Leftrightarrow x=\dfrac{26}{5}\left(TM\right)\)

Vậy tập nghiệm của phương trình \(S=\left\{\dfrac{4}{5};\dfrac{26}{5}\right\}\)

Answer 4

c) \(\left|x-1\right|+\left|x-2\right|+\left|x-3\right|+\left|x-4\right|+30=5x\)

Ta có : \(\left|x-1\right|\ge0\forall x\)

\(\left|x-2\right|\ge0\forall x\\ \left|x-3\right|\ge0\forall x\\ \left|x-4\right|\ge0\forall x\\ \Rightarrow\left|x-1\right|+\left|x-2\right|+\left|x-3\right|+\left|x-4\right|\ge0\forall x\\ \Rightarrow\left|x-1\right|+\left|x-2\right|+\left|x-3\right|+\left|x-4\right|+30\ge30\forall x\\ \Rightarrow5x\ge30\\ \Rightarrow x\ge6\\ \Rightarrow\left\{{}\begin{matrix}x-1\ge5>0\\x-2\ge4>0\\x-3\ge2>0\\x-4\ge1>0\end{matrix}\right.\)

\(pt\Leftrightarrow x-1+x-2+x-3+x-4+30=5x\\ \Leftrightarrow5x=4x+20\\ \Leftrightarrow x=20\)

Vậy nghiệm của phương trình là \(x=20\)