Question 635546
{{{abs(4y-9)>7}}} Start with the given inequality



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


{{{4y-9 < -7}}} or {{{4y-9 > 7}}} Break up the absolute value inequality using the given rule





Now lets focus on the first inequality  {{{4y-9 < -7}}}



{{{4y-9<-7}}} Start with the given inequality



{{{4y<-7+9}}}Add 9 to both sides



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



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




{{{y<1/2}}} Reduce



Now lets focus on the second inequality  {{{4y-9 > 7}}}



{{{4y-9>7}}} Start with the given inequality



{{{4y>7+9}}}Add 9 to both sides



{{{4y>16}}} Combine like terms on the right side



{{{y>(16)/(4)}}} Divide both sides by 4 to isolate y 




{{{y>4}}} Divide




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Answer:


So our answer is


{{{y < 1/2}}} or {{{y > 4}}}



which looks like this in interval notation



*[Tex \LARGE \left(-\infty,\frac{1}{2}\right)\cup\left(4,\infty\right)]



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


{{{drawing(500,50,-10,10,-10,10,
number_line( 500, -7.75, 12.25),

blue(arrow(-2.25,-7,-10,-7)),
blue(arrow(-2.25,-6.5,-10,-6.5)),
blue(arrow(-2.25,-6,-10,-6)),
blue(arrow(-2.25,-5.5,-10,-5.5)),
blue(arrow(-2.25,-5,-10,-5)),
blue(arrow(2.25,-7,10,-7)),
blue(arrow(2.25,-6.5,10,-6.5)),
blue(arrow(2.25,-6,10,-6)),
blue(arrow(2.25,-5.5,10,-5.5)),
blue(arrow(2.25,-5,10,-5)),

circle(-1.75,-5.8,0.35),
circle(-1.75,-5.8,0.4),
circle(-1.75,-5.8,0.45),


circle(1.75,-5.8,0.35),
circle(1.75,-5.8,0.4),
circle(1.75,-5.8,0.45)




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