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已知实数x,y满足|2x+y+1|≤|x+2y+2|,且-1≤y≤1,则z=2x+y的最大值是______.

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已知实数x,y满足|2x+y+1|≤|x+2y+2|,且-1≤y≤1,则z=2x+y的最大值是______.
已知实数x,y满足|2x+y+1|≤|x+2y+2|,且-1≤y≤1,则z=2x+y的最大值是______.
先根据约束条件|2x+y+1|≤|x+2y+2|,-1≤y≤1,化间可得


2x+y+1≥0
x+2y+2≥0
x-y-1≤0,或 ②

2x+y+1≥0
x+2y+2≤0
x+y+1≤0,或 ③

2x+y+1≤0
x+2y+2>0
x+y+1≥0,或④

2x+y+1≤0
x+2y+2≤0
x-y-1≥0.
画出可行域,如图阴影部分,设z=2x+y,
将z的值转化为直线z=2x+y在y轴上的截距,
显然当直线z=2x+y经过点A(2,1)时,z最大,且最大值为5.
故答案为:5.