Question 110280
{{{3(x-4)>7x-9}}} Start with the given inequality




{{{3x-12>7x-9}}} Distribute



{{{3x>7x-9+12}}}Add 12 to both sides



{{{3x-7x>-9+12}}} Subtract 7x from both sides



{{{-4x>-9+12}}} Combine like terms on the left side



{{{-4x>3}}} Combine like terms on the right side



{{{x<(3)/(-4)}}} Divide both sides by -4 to isolate x  (note: Remember, dividing both sides by a negative number flips the inequality sign) 




{{{x<-3 / 4}}} Reduce


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Answer:

So our answer is {{{x<-3 / 4}}} 




Now let's graph the solution set




Set up a number line:

{{{number_line(500,-13,7)}}} 


Now plot the point {{{x=-3/4}}} on the number line



{{{number_line(500,-13,7, -3/4)}}}



Now pick any test point you want, I'm going to choose x=0, and test the inequality {{{x<-3/4}}}



{{{0<-3/4}}} 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=-3/4}}} as the boundary. This means we shade everything to the left of the point {{{x=-3/4}}} like this:

{{{drawing(500,50,-13,7,-10,10,
number_line(500,-13,7),
circle(-3/4,-5.8,0.35),
circle(-3/4,-5.8,0.4),
circle(-3/4,-5.8,0.45),
blue(line(-3/4,-5,-3/4-10,-5)),
blue(line(-3/4,-6,-3/4-10,-6)),
blue(line(-3/4,-7,-3/4-10,-7)),
blue(arrow(-3/4,-5,-3/4-10.2,-5)),
blue(arrow(-3/4,-5.5,-3/4-10.2,-5.5)),
blue(arrow(-3/4,-6,-3/4-10.2,-6))
)}}}  Graph of {{{x<-3/4}}} with the shaded region in blue


note: at the point {{{x=-3/4}}}, there is an <font size=4><b>open</b></font> circle. This means the point {{{x=-3/4}}} is excluded from the solution set.