Question 177246


{{{abs(6-4x)>2}}} Start with the given inequality



Break up the absolute value (remember, if you have {{{abs(x)> a}}}, then {{{x < -a}}} or {{{x > a}}})


{{{6-4x < -2}}} or {{{6-4x > 2}}} Break up the absolute value inequality using the given rule





Now lets focus on the first inequality  {{{6-4x < -2}}}



{{{6-4x<-2}}} Start with the given inequality



{{{-4x<-2-6}}}Subtract 6 from both sides



{{{-4x<-8}}} Combine like terms on the right side



{{{x>(-8)/(-4)}}} Divide both sides by -4 to isolate x  (note: Remember, dividing both sides by a negative number flips the inequality sign) 




{{{x>2}}} Divide



Now lets focus on the second inequality  {{{6-4x > 2}}}



{{{6-4x>2}}} Start with the given inequality



{{{-4x>2-6}}}Subtract 6 from both sides



{{{-4x>-4}}} Combine like terms on the right side



{{{x<(-4)/(-4)}}} Divide both sides by -4 to isolate x  (note: Remember, dividing both sides by a negative number flips the inequality sign) 




{{{x<1}}} Divide




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Answer:


So our answer is


{{{x < 1}}} or {{{x > 2}}} 



which looks like this in interval notation



*[Tex \LARGE \left(-\infty,1\right)\cup\left(2,\infty\right)]



if you wanted to graph the solution set on a number line, you would get:


{{{drawing(500,50,-10,10,-10,10,
number_line( 500, -8.5, 11.5),

blue(arrow(-1,-7,-10,-7)),
blue(arrow(-1,-6.5,-10,-6.5)),
blue(arrow(-1,-6,-10,-6)),
blue(arrow(-1,-5.5,-10,-5.5)),
blue(arrow(-1,-5,-10,-5)),
blue(arrow(1,-7,10,-7)),
blue(arrow(1,-6.5,10,-6.5)),
blue(arrow(1,-6,10,-6)),
blue(arrow(1,-5.5,10,-5.5)),
blue(arrow(1,-5,10,-5)),

circle(-0.5,-5.8,0.35),
circle(-0.5,-5.8,0.4),
circle(-0.5,-5.8,0.45),


circle(0.5,-5.8,0.35),
circle(0.5,-5.8,0.4),
circle(0.5,-5.8,0.45)




)}}} Graph of the solution set in blue and the excluded values represented by open circles