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There are three solutions of the equation ||2x-3|-m|=m have if m>0.
We have to estimate the number of solutions of the equation ||2x - 3| - m| = m where m > 0.
As we know in the modulus function
|x|= x if x>0
-x if x<0
0 if x=0
here given m > 0
if |2x-3|-m>0 then the modulus value will be ||2x-3|-m|= |2x-3|-m
Then, |2x - 3| - m = m
⇒|2x - 3| = m+m= 2m
⇒2x - 3 = 2m or -(2x-3)=2m
⇒2x - 3 = 2m or 2x-3=-2m
⇒2x = 3 ± 2m
⇒x = (3 ± 2m)/2
⇒x= (3 + 2m)/2 or (3 - 2m)/2
If |2x-3|-m<0 then then the modulus value will be | |2x-3|-m| = -(|2x-3|-m)
-(|2x-3|-m)=m
⇒m-|2x-3|=m
⇒-|2x-3|= m-m
⇒ |2x - 3| = 0
⇒2x = 3
⇒x = 3/2
Therefore The solution of the equation will be (3 + 2m)/2, (3 - 2m)/2, and 3/2.
Therefore there are three solutions of the equation ||2x-3|-m|=m have if m>0.
Learn more about the modulus function
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