The solution set of the inequality log21(2x−1)+log21(2−x)≥2log21x is....
Explanation
The logarithmic inequality has special solutions for its conditions
2x−1>0→x>21
2−x>0→x<2
x>0
The special solution is {21<x<2}. The general solution of the inequality is
log21(2x−1)+log21(2−x)≥2log21x
log21((2x−1)(2−x))≥log21x2
log21(−2x2+5x−2)≥log21x2
−2x2+5x−2≤x2
−3x2+5x−2≤0
Multiply by −1, so the sign is flipped
3x2−5x+2≥0
(3x−2)(x−1)≥0
x=32∪x=1
Number line
Number Line Solution
The solution of (3x−2)(x−1)≥0 is x≤32 or x≥1.
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−
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32
1
The solution set is {x≤32∪x≥1}. So the intersection of its solutions
{21<x<2}∩{x≤32∪x≥1}={21<x≤32∪1≤x<2}