Recall that $|x-1|<5$ can be written as $-5

To solve the inequality ∣ x − 1∣ < 5 , we rewrite it as − 5 < x − 1 < 5 and then isolate x , leading to the solution − 4 < x < 6 . This means x can be any value between -4 and 6, excluding -4 and 6. Understanding this concept helps in various practical applications.
;

Answered by Anonymous | 2025-07-04

The given inequality is ∣ x − 1∣ < 5 , which is equivalent to − 5 < x − 1 < 5 .
Add 1 to all sides of the inequality: − 5 + 1 < x − 1 + 1 < 5 + 1 .
Simplify the inequality: − 4 < x < 6 .
The solution set is − 4 < x < 6 , meaning x is between -4 and 6: − 4 < x < 6 ​

Explanation

Understanding the Inequality We are given the inequality ∣ x − 1∣ < 5 , which is equivalent to − 5 < x − 1 < 5 . Our goal is to find the range of values for x that satisfy this inequality.

Isolating x To isolate x , we need to add 1 to all parts of the inequality: − 5 < x − 1 < 5 − 5 + 1 < x − 1 + 1 < 5 + 1 − 4 < x < 6

Finding the Solution Set Therefore, the solution to the inequality ∣ x − 1∣ < 5 is − 4 < x < 6 . This means that x can be any value between -4 and 6, not including -4 and 6.


Examples
Imagine you're setting up a game where players need to guess a number within a certain range. If you tell them the number must be within 5 units of 1, this is mathematically the same as ∣ x − 1∣ < 5 . Solving this inequality, we find that the number must be between -4 and 6. This kind of problem helps in scenarios like setting acceptable error ranges in experiments, defining boundaries in games, or establishing tolerance levels in manufacturing.

Answered by GinnyAnswer | 2025-07-04