Question 115227
#1


{{{(3/5)x<=-54}}} Start with the given inequality




{{{(5)((3/5)x)=(5)(-54)}}} Multiply both sides by 5. 



{{{3x=-270}}} Multiply




{{{x<=(-270)/(3)}}} Divide both sides by 3 to isolate x 




{{{x<=-90}}} Divide


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Answer:

So our answer is {{{x<=-90}}} 




Now let's graph the solution set




Start with the given inequality:


{{{x<=-90}}}


Set up a number line:

{{{number_line(500,-100,-80)}}} 


Now plot the point {{{x=-90}}} on the number line



{{{number_line(500,-100,-80, -90)}}}



Now pick any test point you want, I'm going to choose x=0, and test the inequality {{{x<=-90}}}



{{{0<=-90}}} Plug in {{{x=0}}}



Since this inequality is <font size=4><b>not</b></font> true, we simply shade the entire portion that does <font size=4><b>not</b></font> contain the point x=0 using the point {{{x=-90}}} as the boundary. This means we shade everything to the left of the point {{{x=-90}}} like this:

{{{drawing(500,50,-100,-80,-10,10,
number_line(500,-100,-80, -90),
blue(line(-90,-5,-90-10,-5)),
blue(line(-90,-6,-90-10,-6)),
blue(line(-90,-7,-90-10,-7)),
blue(arrow(-90,-5,-90-10.2,-5)),
blue(arrow(-90,-5.5,-90-10.2,-5.5)),
blue(arrow(-90,-6,-90-10.2,-6))
)}}}  Graph of {{{x<=-90}}} with the shaded region in blue

note: at the point {{{x=-90}}}, there is an <font size=4><b>closed</b></font> circle. This means the point {{{x=-90}}} is included from the solution set.

    

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#2




{{{(-1/3)x<-27}}} Start with the given inequality




{{{(3)((-1/3)x)=(3)(-27)}}} Multiply both sides by 3



{{{-1x=-81}}} Multiply




{{{x>(-81)/(-1)}}} Divide both sides by -1 to isolate x  (note: Remember, dividing both sides by a negative number flips the inequality sign) 




{{{x>81}}} Divide


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

Answer:

So our answer is {{{x>81}}} 



Now let's graph the solution set





Start with the given inequality:


{{{x>81}}}


Set up a number line:

{{{number_line(500,71,91)}}} 


Now plot the point {{{x=81}}} on the number line



{{{number_line(500,71,91, 81)}}}



Now pick any test point you want, I'm going to choose x=0, and test the inequality {{{x>81}}}



{{{0>81}}} Plug in {{{x=0}}}



Since this inequality is <font size=4><b>not</b></font> true, we simply shade the entire portion that does <font size=4><b>not</b></font> contain the point x=0 using the point {{{x=81}}} as the boundary. This means we shade everything to the right of the point {{{x=81}}} like this:

{{{drawing(500,50,71,91,-10,10,
number_line(500,71,91),
circle(81,-5.8,0.35),
circle(81,-5.8,0.4),
circle(81,-5.8,0.45),
blue(line(81,-5,81+10,-5)),
blue(line(81,-6,81+10,-6)),
blue(line(81,-7,81+10,-7)),
blue(arrow(81,-5,81+10.2,-5)),
blue(arrow(81,-5.5,81+10.2,-5.5)),
blue(arrow(81,-6,81+10.2,-6))
)}}}  Graph of {{{x>81}}} with the shaded region in blue

note: at the point {{{x=81}}}, there is an <font size=4><b>open</b></font> circle. This means the point {{{x=81}}} is excluded from the solution set.