Question 165211
To solve inequalities like this, simply solve each inequality individually




Let's solve the first inequality {{{x-1<-1}}}:



{{{x-1<-1}}} Start with the first inequality.



{{{x<-1+1}}} Add {{{1}}} to both sides.



{{{x<0}}} Combine like terms on the right side.



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Now let's solve the second inequality {{{x-5>-1}}}:



{{{x-5>-1}}} Start with the second inequality.



{{{x>-1+5}}} Add {{{5}}} to both sides.



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



So our answer is {{{x<0}}} <font size="4"><b>or</b></font>  {{{x>4}}}





So the solution in interval notation is: <font size="8">(</font>*[Tex \LARGE \bf{-\infty,0}]<font size="8">)</font> *[Tex \LARGE \cup]<font size="8">(</font>*[Tex \LARGE \bf{4,\infty}]<font size="8">)</font>



So the solution in set notation is: *[Tex \LARGE \left\{x\|x<0 \textrm{..or..} x>4\right\}]





Here's the graph of the solution set


{{{drawing(500,80,-5, 9,-10, 10,
number_line( 500, -5, 9 ),


circle(0,0,0.25),
circle(0,0,0.20),


blue(arrow(0,0,-5,0)),
blue(arrow(0,0.30,-5,0.30)),
blue(arrow(0,0.15,-5,0.15)),
blue(arrow(0,-0.15,-5,-0.15)),
blue(arrow(0,-0.30,-5,-0.30)),



circle(4,0,0.25),
circle(4,0,0.20),


blue(arrow(4,0,9,0)),
blue(arrow(4,0.30,9,0.30)),
blue(arrow(4,0.15,9,0.15)),
blue(arrow(4,-0.15,9,-0.15)),
blue(arrow(4,-0.30,9,-0.30))


)}}}



Note:

There is an <b>open</b> circle at {{{x=0}}} which means that we're excluding that value from the solution set.



Also, there is an <b>open</b> circle at {{{x=4}}} which means that we're excluding that value from the solution set.