Respuesta :

JonHenderson55
Answer:  x = 6/5 ; or, write as:  "1  1/5" ; or,write as:  "1.2"  ;

               x =  39/37 = 1  2/37  = 1.0540540540540541 .
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Explanation:
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Given:

9|9−8x|=2x+3 ;  Solve for "x" ;
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DIvide EACH side of the equation by "9" ;
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{ 9|9−8x| } / 9  =  { 2x + 3 } / 9 ;
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to get:
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   |9−8x| = (2x + 3) / 9 ;
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Given the "absolute value" on the left-hand side; we have 
 "Case 1" and "Case 2" scenarios:
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 Case 1:
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      9 − 8x =  (2x+ 3) / 9 ; 

↔   -8x + 9 =  (2x+ 3) / 9 ;
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 Case 2:
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 - (9 − 8x) =  -9 + 8x = 8x − 9 ;

↔     8x − 9 =  (2x + 3) / 9
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Let us start with "Case 1" :
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   -8x + 9 =  (2x + 3) / 9 ;

  → 9*(-8x + 9) = 2x + 3 ;
   
  → -72x + 81  =  2x + 3 ;
 
  → Subtract "81" ; and "2x" from EACH side of the equation ;
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  →  -72x + 81 − 81 − 2x  =  2x + 3 − 81 − 2x ;
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to get:   -74x = -78 ;
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     Divide EACH side of the equation by "-74" ; to isolate "x" on one side of the equation; and to solve for "x" ;
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         -74x / -74  =  -78/-74 ; 
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              x = 78/74 = 39/37 = 1  2/37 ; or, 1.0540540540540541 .
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Now, let's continue with:  "Case 2" :
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  8x − 9 =  (2x + 3) / 9 ;

     →  9*(8x − 9) = 2x + 3 ; 

                →  72x − 81 = 2x + 3 ;
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Subtract "2x" ; and add "81" ; to BOTH sides of the equation:
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                →  72x − 81 − 2x + 81 = 2x + 3 − 2x + 81 ;
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to get:         →   70x  =   84 ;
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        Divide EACH side of the equation by "70" ; to isolate "x" on one side of the equation; and to solve for "x" ;
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                  →    70x / 70 =  84 / 70 ;
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                  →    x = 6/5 =  1  1/5 = 1.2
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BladeRunner212

³⁹/₃₇ and ⁶/₅

Further explanation

For any real number x, the absolute value of x is denoted by | x | and

[tex]\boxed{ \ |x| = \left \{ {{x, \  x \geq 0 } \atop {-x, \ x < 0}} \right. \ }[/tex]

Our case is given by:

[tex]\boxed{ \ 9|9 - 8x| = 2x + 3 \ }[/tex]

Part-1

[tex]\boxed{ \ 9(9 - 8x) = 2x + 3 \ }[/tex]

[tex]\boxed{ \ 81 - 72x = 2x + 3 \ }[/tex]

[tex]\boxed{ \ - 2x - 72x = 3 - 81 \ }[/tex]

[tex]\boxed{ \ - 74x = - 78 \ }[/tex]

[tex]\boxed{ \ x = \frac{-78}{-74} \rightarrow \frac{divided \ by \ (-2)}{divided \ by \ (-2)} \rightarrow \boxed{ \ x = \frac{39}{37} \ } \ }[/tex]

Part-2

[tex]\boxed{ \ - 9(9 - 8x) = 2x + 3 \ }[/tex]

[tex]\boxed{ \ - 81 + 72x = 2x + 3 \ }[/tex]

[tex]\boxed{ \ 72x - 2x = 3 + 81 \ }[/tex]

[tex]\boxed{ \ 70x = 84 \ }[/tex]

[tex]\boxed{ \ x = \frac{84}{70} \rightarrow \frac{divided \ by \ 14}{divided \ by \ 14} \rightarrow \boxed{ \ x = \frac{6}{5} \ } \ }[/tex]

Thus, the solution is [tex]\boxed{ \ x = \frac{39}{37} \ or \ x = \frac{6}{5}. \ }[/tex]

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Keywords: solve the equation, the absolute value, check for extraneous solutions, 9|9 - 8x| = 2x + 3, the solution

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