Question 1093209
.
|x + 24| = -7x.           (1)



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Let me explain how to solve it and how to present the solution in a way &nbsp;<U>how it should be done</U>.



<pre>
As you know, the absolute value term may have different expressions.


1.  If x >= - 24,  then  |x+24| = x + 24,
    
    and the equation &nbsp;(1)&nbsp; takes the form

    x + 24 = -7x.       (2)

    Then  24 = -8x,  x = {{{24/(-8)}}} = -3,

    and, since  x= -3 >= -24,  it is really the solution of  eq(2) in the domain  x >= - 24.

    <U>Therefore</U>, it is the solution to the original equation (1).



2.  If x < -24,  then  |x+24| = -(x + 24),
    
    and the equation &nbsp;(1)&nbsp; takes the form

    -(x + 24) = -7x.     (3)

    Then  -24 = -6x,  x = {{{-24/(-6)}}} = 4.

    <U>BUT</U>  x= 4  is <U>NOT</U> in the domain  x < - 24.

    <U>Therefore</U>,  the value  x= 4  is <U>NOT</U> the solution to the original equation (1).



3.  Thus we found one and only one solution  x = -3.

    The solution  x = 4 was filtered out in the solution process.



4.  The standard way to solve an equation containing the terms under the absolute value sign is to divide the entire number line

    in separate segments, where you can take off the absolute value sign and to write the equation without absolute value sign.

    But then you must remember that the corresponding part of the analysis relates only to this interval of the number line

    which is under consideration.

    If under this local part of the solution you found the root at the considered interval - then it really is the solution 

    to the original absolute value  equation.


    If under this local part of the solution you found the root, but it is not at the considered interval - it still is not 

    the reason to declare this value as the solution to the original absolute value  equation.



5.  By doing in this way, you will not produce EXTRANEOUS solutions at the final stage.

    You will filter them out at the earlier stages of the (local) analysis.
</pre>


See the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/-Absolute-Value-equations.lesson>Absolute Value equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-1.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-2.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 2</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-3.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 3</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Quadratic-Terms-under-Abs-Value-sign-L-1.lesson>HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Quadratic-Terms-under-Abs-Value-sign-L-2.lesson>HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 2</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/absolute-value/Review-of-lessons-on-Absolute-Value-equations.lesson>OVERVIEW of lessons on Absolute Value equations</A> 


Read them attentively and become an expert in this area.
This idea and this methodology were systematically implemented there.
Actually, it is the <U>only way</U> to solve absolute value equations correctly.



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic
"<U>Solving Absolute values equations</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I

https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.