Write an absolute value inequality that represents the given interval.

Question

on a number line from 2 to 14, where 2 and 11 are filled in with two circles, what would an absolute value inequality be? let 'greater than or equal to' and 'less than or equal to' be represented by '<=' and '>='

 

|x (+,-) ___| (<,>,<=,>=) ___

 

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2 3 4 5 6 7 8 9 10 11 12 13 14 

 

sorry if that's confusing, i tried my best to word it simply. /:

 

 

Answer 1

Absolute values and inequalities come together magically if you remember...

1. The midpoint formula: m = (x₂ + x₁)/2

• The midpoint of 2 & 11 are (2 + 11)/2 = 6.5 = m

2. The distance formula: d = |x₂ - x₁|

• The distance from 11 to m is |11-6.5| = 4.5
• The distance from 2 to m is |2-6.5| = 4.5

But we don't only care about x being at 2 or at 11; it can be anywhere in between the two. Take for example x = 4:

•                                         |4-6.5| = 2.5 < 4.5

So notice that any x in between 2 & 11 will have a distance from the midpoint that is less than 4.5.

Therefore, for any x between 2 & 11, and including those numbers should follow the same pattern:

|x-6.5| ≤ 4.5

To emphasize, use the midpoint formula for m and find the farthest possible distance d from that midpoint to get the general formula:

|x-m| ≤ d

Hope this helps!