Solve the equation. check for extraneous solutions. 9|9-8x|=2x+3

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

               x =  39/37 = 1  2/37  = 1.0540540540540541 .

Explanation:

Given:

9|9−8x|=2x+3 ;  Solve for "x" ;

DIvide EACH side of the equation by "9" ;

{ 9|9−8x| } / 9  =  { 2x + 3 } / 9 ;

to get:

   |9−8x| = (2x + 3) / 9 ;

Given the "absolute value" on the left-hand side; we have 
 "Case 1" and "Case 2" scenarios:

 Case 1:

      9 − 8x =  (2x+ 3) / 9 ; 

↔   -8x + 9 =  (2x+ 3) / 9 ;

 Case 2:

 - (9 − 8x) =  -9 + 8x = 8x − 9 ;

↔     8x − 9 =  (2x + 3) / 9

Let us start with "Case 1" :

   -8x + 9 =  (2x + 3) / 9 ;

  → 9*(-8x + 9) = 2x + 3 ;
   
  → -72x + 81  =  2x + 3 ;
 
  → Subtract "81" ; and "2x" from EACH side of the equation ;

  →  -72x + 81 − 81 − 2x  =  2x + 3 − 81 − 2x ;

to get:   -74x = -78 ;
 
     Divide EACH side of the equation by "-74" ; to isolate "x" on one side of the equation; and to solve for "x" ;

         -74x / -74  =  -78/-74 ; 

              x = 78/74 = 39/37 = 1  2/37 ; or, 1.0540540540540541 .

Now, let's continue with:  "Case 2" :

  8x − 9 =  (2x + 3) / 9 ;

     →  9*(8x − 9) = 2x + 3 ; 

                →  72x − 81 = 2x + 3 ;

Subtract "2x" ; and add "81" ; to BOTH sides of the equation:

                →  72x − 81 − 2x + 81 = 2x + 3 − 2x + 81 ;

to get:         →   70x  =   84 ;

        Divide EACH side of the equation by "70" ; to isolate "x" on one side of the equation; and to solve for "x" ;

                  →    70x / 70 =  84 / 70 ;

                  →    x = 6/5 =  1  1/5 = 1.2

³⁹/₃₇ and ⁶/₅

Further explanation

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

Our case is given by:

Part-1

Part-2

Thus, the solution is

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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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