Question 258767
{{{abs(3x + 1) + abs(3 - 2x) = 11}}}
{{{"x" ="?"}}}
<pre><font size = 4 color = "indigo"><b>
We have to solve these 4 cases

Case 1:  {{{""+(3x + 1) +(3 - 2x) = 11}}} 
Case 2:  {{{""+(3x + 1) -(3 - 2x) = 11}}} 
Case 3:  {{{-(3x + 1) + (3 - 2x) = 11}}} 
Case 4:  {{{-(3x + 1) - (3 - 2x) = 11}}} 

and check each in the original to see if it is a
solution or whether it is extraneous.   

Case 1:  

{{{""+(3x + 1) +(3 - 2x) = 11}}}
{{{3x + 1 +3 - 2x = 11}}}
{{{x+4=11}}}
{{{x=7}}}

Check in original:

{{{abs(3x + 1) + abs(3 - 2x) = 11}}}
{{{abs(3(7) + 1) + abs(3 - 2(7)) = 11}}}
{{{abs(21 + 1) + abs(3 - 14) = 11}}}
{{{abs(22) + abs(-11) = 11}}}
{{{22 + 11 = 11}}}
{{{33=11}}}

That does not check. so 7 is extraneous.

Case 2:  

{{{""+(3x + 1) -(3 - 2x) = 11}}}
{{{3x + 1 -3 + 2x = 11}}}
{{{5x-2=11}}}
{{{5x=13}}}
{{{x=13/5}}}


Check in original:

{{{abs(3x + 1) + abs(3 - 2x) = 11}}}
{{{abs(3(13/5) + 1) + abs(3 - 2(13/5)) = 11}}}
{{{abs(39/5 + 1) + abs(3 - 26/5) = 11}}}
{{{abs(39/5+5/5) + abs(15/5-26/5) = 11}}}
{{{abs(44/5) + abs(-11/5) = 11}}}
{{{44/5+11/5=11}}}
{{{55/5=11}}}
{{{11=11}}}

That checks, so {{{13/5}}} is a solution.

Case 3:  

{{{-(3x + 1) +(3 - 2x) = 11}}}
{{{-3x - 1 +3 - 2x = 11}}}
{{{-5x+2=11}}}
{{{-5x=9}}}
{{{x=-9/5}}}

Check in original:

{{{abs(3x + 1) + abs(3 - 2x) = 11}}}
{{{abs(3(-9/5) + 1) + abs(3 - 2(-9/5)) = 11}}}
{{{abs(-27/5 + 1) + abs(3 + 18/5) = 11}}}
{{{abs(-27/5+5/5) + abs(15/5+18/5) = 11}}}
{{{abs(-22/5) + abs(33/5) = 11}}}
{{{22/5+33/5=11}}}
{{{55/5=11}}}
{{{11=11}}}

That checks so {{{-9/5}}} is a solution.

Case 4:  

{{{-(3x + 1) -(3 - 2x) = 11}}}
{{{-3x - 1 -3 + 2x = 11}}}
{{{-x-4=11}}}
{{{-x=15}}}
{{{x=-15}}}

Check in original:

{{{abs(3x + 1) + abs(3 - 2x) = 11}}}
{{{abs(3(-15) + 1) + abs(3 - 2(-15)) = 11}}}
{{{abs(-45 + 1) + abs(3 - 30) = 11}}}
{{{abs(-44) + abs(-27) = 11}}}
{{{44 + 27 = 11}}}
{{{71=11}}}

That does not check, so {{{-15}}} is extraneous.

There are two solutions:

{{{x=13/5}}} and {{{x=-9/5}}}

Edwin</pre>