The inequality log2(x2−x)≤1 has a solution....
Explanation
The condition for the inequality to be satisfied is
x2−x>0
x(x−1)>0
x=0∪x=1
Create its number line
Number Line Condition
The solution of x2−x>0 is x<0 or x>1.
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−
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0
1
So the solution is {x<0∪x>1}. Whereas the general solution of the inequality is
log2(x2−x)≤1
log2(x2−x)≤log22
x2−x≤2
x2−x−2≤0
(x+1)(x−2)≤0
x=−1∪x=2
Second number line
Number Line Solution
The solution of x2−x−2≤0 is −1≤x≤2.
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−
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−1
2
The solution is {−1≤x≤2}. Then the combined solution is
{x<0∪x>1}∩{−1≤x≤2}={−1≤x<0∪1<x≤2}