Question 1124511
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how many solution does the equation a|x+b|+c=d have if a < 0 and c = d? If a < 0, d > 0 and c < d ? Explain
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(a) &nbsp;&nbsp;How many solution does the equation a|x+b|+c=d have if a < 0 and c = d ?


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An equation

    a|x+b| + c = d   

is equivalent to

    a|x+b| = d - c.


If c = d, then this equation becomes

    a|x+b| = 0.


If a =/= 0  (as it is given in our case), then the last equation implies

   |x+b| = 0,

which has one and only one unique solution  x= -b.


<U>Answer</U>.  In this case the original equation has a unique solution.
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(b) &nbsp;&nbsp;How many solution does the equation a|x+b|+c=d have if a < 0, d > 0 and c < d ?


<pre>
An equation

    a|x+b| + c = d   

is equivalent to

    a|x+b| = d - c.     (*)


If c < d, then the right side of the last equation is positive, while its left side is negative.


Therefore, the equality (*) is not possible.

Or, in other words, the equation (*) has no solutions.


<U>Answer</U>.  Under the given condition, the given equation has no solution.
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Solved, answered and explained.