Question

Write the solutions as either the union or the intersection of two sets.
What are the solutions of |-x +10|<5?
[]Will medal and fan.[]

Answer 1

There are a number of ways to solve this.

I'll teach you the standard approach.

Definition:
\[|a| = a ~if~a \ge 0\]
\[|a| = -a ~if ~ a < 0\]

So we write out |-x+10|< 5 as two cases based on positive/negative conditions.
Case 1: |-x+10| = -x + 10 < 5 whenever -x +10 > 0
Case 2: |-x+10| = -(-x + 10) < 5 whenever -x + 10 < 0

This simplifies to
Case 1: - x + 10 < 5 whenever x < 10 --> -x < -5 so x > 5
Case 2: -(-x+10) < 5 whenever x > 10 --> x - 10 < 5 so x < 15

Thus the conditions suggest
Case 1 5 < x < 10 or Case 2 10 < x < 15 which combines to saying 5 < x < 15

The shortcut to this method is that if you do a lot of these problems you notice that
|x - a| < k simplifies to - k < x-a < k (which you simplify further)
and
|x-a| > k simplifies to x- a < -k or x -a > k
Thus |-x+10|< 5 is just - 5 < -x+10 < 5 so -15 < -x < -5 so 5 < x < 15.

Graphical method. |-x+10| < 5 is the same as asking what value when I take the distance from 10, is less than 5 units away from 10?


Thus 5 and 15 wedge in a distance of 5 unit from 10.