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

The inequality ∣ x − 1∣ < 5 can be rewritten as − 5 < x − 1 < 5 , which simplifies to − 4 < x < 6 . This means that x can take any value between -4 and 6. Thus, the solution set is − 4 < x < 6 ​ .
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Answered by Anonymous | 2025-07-04

The given inequality is ∣ x − 1∣ < 5 , which can be rewritten as − 5 < x − 1 < 5 .
Add 1 to all parts of the inequality to isolate x : − 5 + 1 < x − 1 + 1 < 5 + 1 .
Simplify the inequality: − 4 < x < 6 .
The solution set for x is all values 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

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


Examples
Understanding absolute value inequalities is crucial in various fields. For instance, in engineering, when designing a system, you might need to ensure that a certain parameter (like temperature or pressure) stays within a specific range around a target value. If the target temperature is T 0 ​ and the acceptable range is ± Δ T , then the temperature T must satisfy ∣ T − T 0 ​ ∣ < Δ T . This ensures the system operates safely and efficiently. Similarly, in finance, you might use absolute value inequalities to model risk, ensuring that investment returns stay within acceptable bounds.

Answered by GinnyAnswer | 2025-07-04