Question 151733

Let's solve the first inequality {{{2x>-6}}}:



{{{2x>-6}}} Start with the first inequality.



{{{x>(-6)/(2)}}} Divide both sides by {{{2}}} to isolate {{{x}}}. 



{{{x>-3}}} Reduce.



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Now let's solve the second inequality {{{x-4<3}}}:



{{{x-4<3}}} Start with the second inequality.



{{{x<3+4}}} Add {{{4}}} to both sides.



{{{x<7}}} Combine like terms on the right side.



So our answer is {{{x>-3}}} <font size="4"><b>and</b></font>  {{{x<7}}}



This combines to the compound inequality {{{-3<x<7}}}



So the answer in interval notation is   <font size="8">(</font>*[Tex \LARGE \bf{-3,7}]<font size="8">)</font>



So the answer in set-builder notation is  *[Tex \LARGE \left\{x\|-3 < x < 7\right\}]



Here's the graph of the solution set


{{{drawing(500,80,-8, 12,-10, 10,
number_line( 500, -8, 12 ),

blue(line(-3,0,7,0)),
blue(line(-3,0.30,7,0.30)),
blue(line(-3,0.15,7,0.15)),
blue(line(-3,-0.15,7,-0.15)),
blue(line(-3,-0.30,7,-0.30)),
circle(-3,0,0.25),circle(-3,0,0.20),
circle(7,0,0.25),
circle(7,0,0.20)

)}}} Graph of the solution set


Note:

There is an <b>open</b> circle at {{{x=-3}}} which means that we're excluding this value from the solution set

Also, there is an <b>open</b> circle at {{{x=7}}} which means that we're excluding this value from the solution set.