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What interval consists of all $w$ which satisfy ${\bf neither}$ $-2(6+2w)\le -16$ nor $-3w\ge 18$ ?

 Jul 13, 2019
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What interval consists of all \(w\) which satisfy \(\bf neither\) \(-2(6+2w)\le -16\) nor \(-3w\ge 18\) ?

 

 

If  w  makes the inequality   -2(6 + 2w) ≤ -16   false, then

it must make the inequality  -2(6 + 2w) > -16   true.


-2(6 + 2w)  >  -16

                             Divide both sides by  -2, a negative number, so flip the sign

6 + 2w  <  8

                             Subtract  6  from both sides of the inequality.

2w  <  2

                             Divide both sides by  2, a positive number, so don't flip the sign

w  <  1

 

If   -2(6 + 2w) > -16   then   w < 1

 

 

If  w  makes the inequality   -3w ≥ 18   false, then

it must make the inequality  -3w < 18   true.

 

-3w  <  18

                  Divide both sides by  3, a negative number, so flip the sign.

w  >  -6

 

If   -3w < 18   then   w > -6

 

 

So the solution to the question is all  w  such that   w < 1   and   w > -6

That is all  w  in the interval   (-6, 1)

 Jul 13, 2019

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