Question 166106
{{{abs(-3x-5)>=10}}} Start with the given inequality



Break up the absolute value (remember, if you have {{{abs(x)>= a}}}, then {{{x <= -a}}} or {{{x >= a}}})


{{{-3x-5 <= -10}}} or {{{-3x-5 >= 10}}} Break up the absolute value inequality using the given rule





Now lets focus on the first inequality  {{{-3x-5 <= -10}}}



{{{-3x-5<=-10}}} Start with the given inequality



{{{-3x<=-10+5}}}Add 5 to both sides



{{{-3x<=-5}}} Combine like terms on the right side



{{{x>=(-5)/(-3)}}} Divide both sides by -3 to isolate x  (note: Remember, dividing both sides by a negative number flips the inequality sign) 




{{{x>=5/3}}} Reduce



Now lets focus on the second inequality  {{{-3x-5 >= 10}}}



{{{-3x-5>=10}}} Start with the given inequality



{{{-3x>=10+5}}}Add 5 to both sides



{{{-3x>=15}}} Combine like terms on the right side



{{{x<=(15)/(-3)}}} Divide both sides by -3 to isolate x  (note: Remember, dividing both sides by a negative number flips the inequality sign) 




{{{x<=-5}}} Divide




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Answer:


So our answer is


{{{x >= 5/3}}} or {{{x <= -5}}}



So the solution in interval notation is: <font size="8">(</font>*[Tex \LARGE \bf{-\infty,-5}]<font size="8">]</font> *[Tex \LARGE \cup]<font size="8">[</font>*[Tex \LARGE \bf{\frac{5}{3},\infty}]<font size="8">)</font>



So the solution in set builder notation is: *[Tex \LARGE \left\{x\|x\le-5 \textrm{..or..} x\ge\frac{5}{3}\right\}]





Here's the graph of the solution set


{{{drawing(500,80,-10, 6.66666666666667,-10, 10,
number_line( 500, -10, 6.66666666666667 ,-5,5/3),

blue(arrow(-5,0,-10,0)),
blue(arrow(-5,0.30,-10,0.30)),
blue(arrow(-5,0.15,-10,0.15)),
blue(arrow(-5,-0.15,-10,-0.15)),
blue(arrow(-5,-0.30,-10,-0.30)),


blue(arrow(5/3,0,6.66666666666667,0)),
blue(arrow(5/3,0.30,6.66666666666667,0.30)),
blue(arrow(5/3,0.15,6.66666666666667,0.15)),
blue(arrow(5/3,-0.15,6.66666666666667,-0.15)),
blue(arrow(5/3,-0.30,6.66666666666667,-0.30))


)}}}



Note:

There is a <b>closed</b> circle at {{{x=-5}}} which means that we're including that value from the solution set.



Also, there is a <b>closed</b> circle at {{{x=5/3}}} which means that we're including that value from the solution set.