Question 107469
#1


{{{abs(5x+6)>26}}} Start with the given inequality



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


{{{5x+6 < -26}}} or {{{5x+6 > 26}}} Break up the absolute value inequality using the given rule





Now lets focus on the first inequality  {{{5x+6 < -26}}}



{{{5x+6<-26}}} Start with the given inequality



{{{5x<-26-6}}}Subtract 6 from both sides



{{{5x<-32}}} Combine like terms on the right side



{{{x<(-32)/(5)}}} Divide both sides by 5 to isolate x 




{{{x<-32/5}}} Reduce



Now lets focus on the second inequality  {{{5x+6 > 26}}}



{{{5x+6>26}}} Start with the given inequality



{{{5x>26-6}}}Subtract 6 from both sides



{{{5x>20}}} Combine like terms on the right side



{{{x>(20)/(5)}}} Divide both sides by 5 to isolate x 




{{{x>4}}} Divide




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


Answer:


So our answer is


{{{x < -32/5}}} or {{{x > 4}}}



which looks like this in interval notation



*[Tex \LARGE \left(-\infty,\frac{-32}{5}\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, -11.2, 8.8),

blue(arrow(-5.7,-7,-10,-7)),
blue(arrow(-5.7,-6.5,-10,-6.5)),
blue(arrow(-5.7,-6,-10,-6)),
blue(arrow(-5.7,-5.5,-10,-5.5)),
blue(arrow(-5.7,-5,-10,-5)),
blue(arrow(5.7,-7,10,-7)),
blue(arrow(5.7,-6.5,10,-6.5)),
blue(arrow(5.7,-6,10,-6)),
blue(arrow(5.7,-5.5,10,-5.5)),
blue(arrow(5.7,-5,10,-5)),

circle(-5.2,-5.8,0.35),
circle(-5.2,-5.8,0.4),
circle(-5.2,-5.8,0.45),


circle(5.2,-5.8,0.35),
circle(5.2,-5.8,0.4),
circle(5.2,-5.8,0.45)




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



<hr>


#2




{{{abs(2x-4)=16}}} Start with the given equation



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


{{{2x-4=-16}}} or {{{2x-4=16}}} Set the expression {{{2x-4}}} equal to the original value 16 and it's opposite -16





Now lets focus on the first equation  {{{2x-4=-16}}}



{{{2x-4=-16}}} Start with the given equation



{{{2x=-16+4}}}Add 4 to both sides



{{{2x=-12}}} Combine like terms on the right side



{{{x=(-12)/(2)}}} Divide both sides by 2 to isolate x




{{{x=-6}}} Divide


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

Answer:

So our answer is {{{x=-6}}} 



<hr>


Now lets focus on the second equation {{{2x-4=16}}}




{{{2x-4=16}}} Start with the given equation



{{{2x=16+4}}}Add 4 to both sides



{{{2x=20}}} Combine like terms on the right side



{{{x=(20)/(2)}}} Divide both sides by 2 to isolate x




{{{x=10}}} Divide


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

Answer:

So our answer is {{{x=10}}} 




So the solutions to {{{abs(2x-4)=16}}} are:


{{{x=-6}}} and {{{x=10}}}




Notice if we graph  {{{y=abs(2x-4)}}} and {{{y=16}}} (just set each side equal to y and graph), we get



{{{graph(500,500,-20,20,-2,20,abs(2x-4),16)}}}  Graph of {{{y=abs(2x-4)}}} (red) and {{{y=16}}}(green)


and we can see the two graphs intersect at {{{x=-6}}} and {{{x=10}}}. So this verifies our answer.