Express the fact that #x # differs from 2 by less than#1/ 2# as an inequality involving an absolute value. What is #x#?

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Answer 1 · 2017-09-17T14:48:10

#|x - 2| < frac(1)(2)#

#frac(3)(2) < x < frac(5)(2)#

What the statement is saying is that the magnitude of the difference between #x# and #2# will be less than #frac(1)(2)#.

The difference between #x# and #2# can be expressed as #x - 2#.

The magnitude of any value or expression can be found by using absolute value.

So the magnitude of the difference can be expressed as #|x - 2|#.

Now, this whole expression is less than #frac(1)(2)#, or #|x - 2| < frac(1)(2)#.

Let's solve this inequality using the properties of absolute value:

#Rightarrow x - 2 < frac(1)(2) Rightarrow x < frac(1)(2) + 2 therefore x < frac(5)(2)#

#or#

#Rightarrow x - 2 > - frac(1)(2) Rightarrow x > 2 - frac(1)(2) therefore x > frac(3)(2)#

If we combine these two results, we get the set #frac(3)(2) < x < frac(5)(2)#.

Therefore, #x# is a number between, but not including, #frac(3)(2)# and #frac(5)(2)#.