How do you solve and write the following in interval notation: # abs(x-5)<=7#?

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Answer 1 · 2016-07-12T15:06:09

#-2<=x<=12.#

In the Interval Notation, we write this as, #x in [-2,12].#

Recall the Defn. of the Absolute Value of #t in RR : |t|=t, if t>=0, &, |t|=-t, if t<0.#

So, we need to consider two cases #: (i) (x-5)>=0, (ii) (x-5)<0#

Case #(i) : (x-5)>=0.#
#:. |x-5|=x-5#, & hence,
#|x-5|<=7 rArr x-5<=7rArr x<=5+7=12.#

Case #(ii) : (x-5)<0.#
#:. |x-5|=-(x-5)=5-x#, so that,
#|x-5|<=7 rArr 5-x<=7 rArr-2<=x.#

Combining these cases, we have, #-2<=x<=12.#

In the Interval Notation, we write this as, #x in [-2,12].#