The solutions to the inequalities are -13 < x < 15 and x > 15 and x < -13
The inequality expressions are given as:
|x – 1| + 1 < 15
|x – 1| + 1 > 15
We start by solving the inequality expression |x – 1| + 1 > 15
So, we have
|x – 1| + 1 < 15
Subtract 1 from both sides
|x – 1| < 14
Remove the absolute bracket
x – 1 < 14 or x - 1 > -14
This gives
x < 15 and x > -13
Combine the inequalities
-13 < x < 15
We then continue by solving the inequality expression |x – 1| + 1 < 15
So, we have
|x – 1| + 1 > 15
Subtract 1 from both sides
|x – 1| > 14
Remove the absolute bracket
x – 1 > 14 or x - 1 < -14
This gives
x > 15 and x < -13
Hence, the solutions to the inequalities are -13 < x < 15 and x > 15 and x < -13
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