Question

CMR :

a, \(\left|x-1\right|+\left|x-3\right|+\left|x-5\right|+\left|x-7\right|\ge8\)

b, \(\left|x+1\right|+\left|x+3\right|+\left|x+5\right|\ge4\)

c, \(\left|x-1\right|+2\left|x-3\right|+\left|x-5\right|\ge4\)

Cần gấp lời giải đầy đủ dễ hiểu và đáp án nhé

Answer 1

\(\left|x-1\right|+\left|x-3\right|+\left|x-5\right|+\left|x-7\right|=\left(\left|x-1\right|+\left|x-7\right|\right)+\left(\left|x-3\right|+\left|x-5\right|\right)\\ \)

\(=\left(\left|x-1\right|+\left|7-x\right|\right)+\left(\left|x-3\right|+\left|5-x\right|\right)\)

\(\ge\left|x-1+7-x\right|+\left|x-3+5-x\right|=\left|6\right|+\left|2\right|=8\)

Answer 2

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

\(\ge\left|x+1+3-x\right|+\left|x+5\right|=\left|4\right|+\left|x+5\right|=4+\left|x+5\right|\ge4\)

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

\(\ge\left|x-1+5-x\right|+2\left|x-3\right|=\left|4\right|+2\left|x-3\right|=4+2\left|x-3\right|\ge4\)