Problem: How many solutions does the equation \[ ||2x-3|-m|=m \] have if m>0 ?
Problem: How many solutions does the equation \[ ||2x-3|-m|=m \] have if m>0 ?
Solution:
It is given that ‘m’ is positive, so the LHS of the equation is positive !
Hence we can write,
\[ |2x-3|-m=m \]
or
\[ |2x-3|=2m \]
or
\[ 2x-3=\pm2m \]
or
\[ 2x=(\pm 2m+3) \]
or
\[ x=\frac{\pm2m+3}{2} \]
Which expresses two solutions so for !
But
\[ |2x-3|=0 \]
as well,
So,
from there we can say,
\[ x=\frac{3}{2} \]
is the thired solution,
So the equation has three solutions !
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