Question 119350
#1





{{{abs(2t+5)<7}}} Start with the given inequality



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


{{{2t+5 > -7}}} and {{{2t+5 < 7}}} Break up the absolute value inequality using the given rule



{{{-7 < 2t+5 < 7}}} Combine the two inequalities to get a compound inequality




{{{-12 < 2t < 2}}} Subtract 5 from  all sides



{{{-6 < t < 1}}}  Divide all sides by 2 to isolate t




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


Answer:


So our answer is


{{{-6 < t < 1}}}




which looks like this in interval notation



*[Tex \LARGE \left(-6,1\right)]



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


{{{drawing(500,50,-10,10,-10,10,
number_line( 500, -12.5, 7.5),

blue(line(-3,-7,3.15,-7)),
blue(line(-3,-6,3.15,-6)),
blue(line(-3,-5,3.15,-5)),

circle(-3.5,-5.8,0.35),
circle(-3.5,-5.8,0.4),
circle(-3.5,-5.8,0.45),


circle(3.5,-5.8,0.35),
circle(3.5,-5.8,0.4),
circle(3.5,-5.8,0.45)




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




<hr>



#2




{{{abs(w-5)>3}}} Start with the given inequality



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


{{{w-5 < -3}}} or {{{w-5 > 3}}} Break up the absolute value inequality using the given rule





Now lets focus on the first inequality  {{{w-5 < -3}}}



{{{w-5<-3}}} Start with the given inequality



{{{w<-3+5}}}Add 5 to both sides



{{{w<2}}} Combine like terms on the right side



Now lets focus on the second inequality  {{{w-5 > 3}}}



{{{w-5>3}}} Start with the given inequality



{{{w>3+5}}}Add 5 to both sides



{{{w>8}}} Combine like terms on the right side




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


Answer:


So our answer is


{{{w < 2}}} or {{{w > 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, -5, 15),

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