Question 319141


{{{abs(2a-6)>10}}} Start with the given inequality



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


{{{2a-6 < -10}}} or {{{2a-6 > 10}}} Break up the absolute value inequality using the given rule





Now lets focus on the first inequality  {{{2a-6 < -10}}}



{{{2a-6<-10}}} Start with the given inequality



{{{2a<-10+6}}}Add 6 to both sides



{{{2a<-4}}} Combine like terms on the right side



{{{a<(-4)/(2)}}} Divide both sides by 2 to isolate a 




{{{a<-2}}} Divide



Now lets focus on the second inequality  {{{2a-6 > 10}}}



{{{2a-6>10}}} Start with the given inequality



{{{2a>10+6}}}Add 6 to both sides



{{{2a>16}}} Combine like terms on the right side



{{{a>(16)/(2)}}} Divide both sides by 2 to isolate a 




{{{a>8}}} Divide




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


Answer:


So our answer is


{{{a < -2}}} or {{{a > 8}}}



which looks like this in interval notation



*[Tex \LARGE \left(-\infty,-2\right)\cup\left(8,\infty\right)]



if you wanted to graph the solution set, you would get


{{{drawing(500,50,-10,10,-10,10,
number_line( 500, -7, 13),

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

circle(-5,-5.8,0.35),
circle(-5,-5.8,0.4),
circle(-5,-5.8,0.45),


circle(5,-5.8,0.35),
circle(5,-5.8,0.4),
circle(5,-5.8,0.45)




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