设命题p:实数x满足x2-4ax+3a2<0(a>0),命题q:实数x满足x−3x−2≤0,
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设命题p:实数x满足x2-4ax+3a2<0(a>0),命题q:实数x满足
x−3 |
x−2 |
(1)若a=1,解x2-4x+3<0得:1<x<3,解
x−3
x−2≤0得:2<x≤3;
∴命题p:实数x满足1<x<3,命题q:实数x满足2<x≤3;
∵p∧q为真,∴p真,q真,∴x应满足
1<x<3
2<x≤3,解得2<x<3,即x的取值范围为(2,3);
(2)¬q为:实数x满足x<2,或x>3;¬p为:实数x满足x2-4ax+3a2≥0,并解x2-4ax+3a2≥0得x≤a,或x≥3a;
¬p是¬q的充分不必要条件,所以a应满足:a<2,且3a>3,解得1<a<2;
∴a的取值范围为:(1,2).
x−3
x−2≤0得:2<x≤3;
∴命题p:实数x满足1<x<3,命题q:实数x满足2<x≤3;
∵p∧q为真,∴p真,q真,∴x应满足
1<x<3
2<x≤3,解得2<x<3,即x的取值范围为(2,3);
(2)¬q为:实数x满足x<2,或x>3;¬p为:实数x满足x2-4ax+3a2≥0,并解x2-4ax+3a2≥0得x≤a,或x≥3a;
¬p是¬q的充分不必要条件,所以a应满足:a<2,且3a>3,解得1<a<2;
∴a的取值范围为:(1,2).
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