Question 1209411
Before graphing calculators, it would have been a more complicated
 
The right hand is greater than 30 except for all values of {{{x}}}
{{{30+abs(6x-2)+abs(6x-2)>=30}}} for all of real values because
{{{abs(6x-2)>=0}}} for all real numbers,
with {{{abs(6x-2)=6x-2>=0}}} for {{{x>=1/3}}}, and {{{abs(6x-2)=-6x+2}}} otherwise.
{{{abs(x^2 - 20x - 4)>=0}}} for
{{{x<=10-sqrt(104)=approximately}}}{{{-0.198039027}}} and {{{x>=10+sqrt(104)=approximately}}}{{{20.198039027}}}.
{{{x^2 - 20x - 4<0}}} and {{{abs(x^2 - 20x - 4)>0}}} for
for {{{x}}} such that {{{10-sqrt(104)<x<10+sqrt(104)}}}
 
For the left hand side, {{{x^2 - 30x - 1 =0}}}, has as solutions {{{x=15 +- sqrt(226)}}}
For values of {{{x<=15 - sqrt(226)=approximately}}}{{{-0.033296378}}} and {{{x>=15 + sqrt(226)=approximately}}}{{{30.033296378}}},
{{{x^2 - 30x - 1>=0}}} and {{{abs(x^2 - 30x - 1)=x^2 - 30x - 1}}}.
For {{{15 - sqrt(226)<x<15 + sqrt(226)}}} {{{abs(x^2 - 30x - 1)=-x^2 + 30x + 1}}}
Those limiting values are approximately {{{-0.033296378}}} and {{{30.033296378}}}.

 
For {{{x<=10-sqrt(104)=approximately}}}{{{-0.198039027}}}, {{{6x-2<0}}}, while {{{x^2 - 30x - 1 >0}}} and {{{x^2 - 20x - 4>=0}}}
and the equation turns into
{{{x^2 - 30x - 1 = -6x+2 + x^2 - 20x - 4}}}
{{{-30x + 6x + 20x = 30 + 2 - 4 + 1}}}
{{{-4x = 29}}} --> {{{x=29/(-4)=highlight(-7.25)}}}
 
For {{{-1<x<1}}} {{{x^2 - 30x - 1}}} is decreasing from {{{30}}} at {{{x=-1}}} to a minimum 0f {{{-1}}} at {{{x=0}}} and then increasing to {{{30}}} at {{{x=1}}} and {{{abs(x^2 - 30x - 1)}}} is less than the value of {{{30+abs(6x-2)+abs(x^2 - 20x - 4)>=30}}}
 =
For {{{1<=x<=10+sqrt(104)}}}, {{{abs(6x-2)=6x-2>0}}}, {{{x^2-20x-4<=0}}}, and {{{x^2-30x-1<0}}},
so {{{abs(x^2-20x-4)=-x^2+20x+4)}}} and {{{abs(x^2-30x-1)=-x^2+30x+1}}}
The equation turns into
{{{-x^2+30x+1=30+6x-2+-x^2+20x+4}}}
{{{30x-6x-20x=30-2+4-1}}}
{{{4x=31}}} --> {{{x=31/4=highlight(7.75)}}}
 
For {{{10+sqrt(104)<x<15+sqrt(226)}}}  {{{abs(6x-2)=6x-2>0}}} while
{{{abs(x^2-20x-4)=x^2-20x-4>0}}} and {{{x^2-30x-1<0}}}, so {{{abs(x^2-30x-1)=-x^2+30x+1}}}
turning the equation into {{{-x^2+30x+1=30+6x-2+x^2-20x-4}}}
{{{0=x^2-30x-1+30+6x-2+x^2-20x-4}}}
{{{0=2x^2-44x+23}}} --> {{{x=(44 +- sqrt(1752))/4=11 +- sqrt(438)/2}}}
{{{x=11-sqrt(438)/2=approximately}}}{{{0.536}}} does not comply with {{{10+sqrt(104)<x<15+sqrt(226)}}}
{{{x=highlight(11+sqrt(438)/2)=approximately}}}{{{21.464225}}} is a solution between {{{10+sqrt(104)=approximately}}}{{{20.198039027}}} and {{{15+sqrt(226)=approximately}}}{{{30.033296378}}}
 
For {{{x>=15+sqrt(226)}}} {{{abs(6x-2)=6x-2>0}}}, {{{abs(x^2-20x-4)=x^2-20x-4>0}}} , and {{{abs(x^2-30x-1)=x^2-30x-1>=0}}} ,
turning the equation into {{{x^2-30x-1=30+6x-2+x^2-20x-4}}}
{{{-30x-6x+20x=30-2-4+1}}}
{{{-16x=25}}} --> {{{x=25/(-16)=-1.5625}}} is not a solution complying with {{{x>=15+sqrt(226)=approximately}}}{{{30.033296378}}}