Question 174988
{{{abs(x-4)>3}}} Start with the given inequality



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


{{{x-4 < -3}}} or {{{x-4 > 3}}} Break up the absolute value inequality using the given rule





Now lets focus on the first inequality  {{{x-4 < -3}}}



{{{x-4<-3}}} Start with the given inequality



{{{x<-3+4}}}Add 4 to both sides



{{{x<1}}} Combine like terms on the right side



Now lets focus on the second inequality  {{{x-4 > 3}}}



{{{x-4>3}}} Start with the given inequality



{{{x>3+4}}}Add 4 to both sides



{{{x>7}}} Combine like terms on the right side




----------------------------------------------------


Answer:


So our answer is


{{{x < 1}}} or {{{x > 7}}}



which looks like this in interval notation



*[Tex \LARGE \left(-\infty,1\right)\cup\left(7,\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, -6, 14),

blue(arrow(-3.5,-7,-10,-7)),
blue(arrow(-3.5,-6.5,-10,-6.5)),
blue(arrow(-3.5,-6,-10,-6)),
blue(arrow(-3.5,-5.5,-10,-5.5)),
blue(arrow(-3.5,-5,-10,-5)),
blue(arrow(3.5,-7,10,-7)),
blue(arrow(3.5,-6.5,10,-6.5)),
blue(arrow(3.5,-6,10,-6)),
blue(arrow(3.5,-5.5,10,-5.5)),
blue(arrow(3.5,-5,10,-5)),

circle(-3,-5.8,0.35),
circle(-3,-5.8,0.4),
circle(-3,-5.8,0.45),


circle(3,-5.8,0.35),
circle(3,-5.8,0.4),
circle(3,-5.8,0.45)




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