Question 1208247
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|x+4| + 2|x+4| + 4|x+4| + 5 = 19.
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<pre>
It is easy.


On the left side, you have  three addends with identical part |x+4|.

So, you combine like terms in the left side. Then your equation takes the form

    7*|x+4| + 5 = 19.


Now move constant term 5 from left side to the right (by subtracting 5 from both sides).

You get then

    7*|x+4| = 19 - 5,

or

    7*|x+4| = 14.


Next, you divide both sides by 7.  It gives you

    |x+4| = 14/7 = 2.


Thus, the absolute value of (x+4) is 2.  It means that x+4 is EITHER 2 OR -2.

Therefore, we consider two cases.


    Case 1.  x+4 = 2.  Subtract 4 from both sides.  You will get

             x = 2 - 4 = -2.  So, x = -2 is the solution.


    Case 2.  x+4 = -2.  Again, subtract 4 from both sides.  You will get

             x = -2 - 4 = -6.  So, x = -6 is another solution.


Thus, the given equation has two solutions:  x= -2  and x= -6.


<U>CHECK</U>.  First, check that that x= -2 is the solution.

        You have  |-2+4| = |2| = 2.  ! correct !


        Next, check that x= -6 is the solution.  

        You have  |-6+4| = |-2| = 2.  ! correct !
</pre>

At this point, the solution is complete.


Come again to this forum soon to learn something new !


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