Question 168704


{{{x-1<=3x+1<=x+5}}} Start with the given compound inequality.



Break up the compound inequality to get:



{{{x-1<=3x+1}}} AND {{{3x+1<=x+5}}}



So let's solve the first inequality {{{x-1<=3x+1}}}



{{{x-1<=3x+1}}} Start with the given inequality.



{{{x<=3x+1+1}}} Add {{{1}}} to both sides.



{{{x-3x<=1+1}}} Subtract {{{3x}}} from both sides.



{{{-2x<=1+1}}} Combine like terms on the left side.



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



{{{x>=(2)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{x}}}. note: Remember, the inequality sign flips when we divide both sides by a negative number. 



{{{x>=-1}}} Reduce.



Now let's solve the second inequality {{{3x+1<=x+5}}}



{{{3x+1<=x+5}}} Start with the given inequality.



{{{3x<=x+5-1}}} Subtract {{{1}}} from both sides.



{{{3x-x<=5-1}}} Subtract {{{x}}} from both sides.



{{{2x<=5-1}}} Combine like terms on the left side.



{{{2x<=4}}} Combine like terms on the right side.



{{{x<=(4)/(2)}}} Divide both sides by {{{2}}} to isolate {{{x}}}. 



{{{x<=2}}} Reduce.


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Answer:


So the solution is {{{x>=-1}}} <b>and</b> {{{x<=2}}}.



Our answer also looks like {{{-1<=x<=2}}}.






So the answer in interval notation is   <font size="8">[</font>*[Tex \LARGE \bf{-1,2}]<font size="8">]</font>



Also, the answer in set-builder notation is  *[Tex \LARGE \left\{x\|-1 \le x \le 2\right\}]



Here's the graph of the solution set


{{{drawing(500,80,-6, 7,-10, 10,
number_line( 500, -6, 7 ,-1,2),

blue(line(-1,0,2,0)),
blue(line(-1,0.30,2,0.30)),
blue(line(-1,0.15,2,0.15)),
blue(line(-1,-0.15,2,-0.15)),
blue(line(-1,-0.30,2,-0.30))

)}}} Graph of the solution set


Note:

There is a <b>closed</b> circle at {{{x=-1}}} which means that we're including this value in the solution set

Also, there is a <b>closed</b> circle at {{{x=2}}} which means that we're including this value in the solution set.