The solution set of the inequality ∣x−5∣2−3∣x−5∣+2<0 is....
Explanation
Let ∣x−5∣=p, we can solve the inequality to become
∣x−5∣2−3∣x−5∣+2<0
p2−3p+2<0
(p−1)(p−2)<0
p=1∪p=2
Return to the original form
p=1→∣x−5∣=1→x=6∪x=4
p=2→∣x−5∣=2→x=7∪x=3
The number line is as follows
Number Line Solution
The solution of (p−1)(p−2)<0 for p=∣x−5∣ yields 3<x<4 or 6<x<7.
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−
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−
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3
4
4
6
6
7
Thus it has two solutions, namely HP={3<x<4}∪{6<x<7}. If shortened to (3,4)∪(6,7).