Which of the following is equivalent to [tex]$|x-1|<5$[/tex]?
A. [tex]$x-1<5$[/tex]
B. [tex]$5 x-1<5$[/tex]
Answer (2)
The equivalent inequality for ∣ x − 1∣ < 5 is − 5 < x − 1 < 5 . Therefore, the correct choice is option C. This conveys that the distance of x − 1 from 0 is less than 5, resulting in x being bounded between -4 and 6.
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Recall the definition of absolute value: ∣ a ∣ < b is equivalent to − b < a < b .
Apply this definition to the given inequality ∣ x − 1∣ < 5 , which gives − 5 < x − 1 < 5 .
Compare this result with the given options.
The equivalent inequality is − 5 < x − 1 < 5 .
Explanation
Understanding the Problem We are given the inequality ∣ x − 1∣ < 5 and asked to find an equivalent inequality from the given options.
Definition of Absolute Value Recall the definition of absolute value: ∣ a ∣ < b is equivalent to − b < a < b . This means that the distance between a and 0 is less than b .
Applying the Definition Applying this definition to the given inequality ∣ x − 1∣ < 5 , we get − 5 < x − 1 < 5 . This means that the distance between x − 1 and 0 is less than 5.
Finding the Equivalent Inequality Comparing this result with the given options, we see that the equivalent inequality is − 5 < x − 1 < 5 .
Final Answer Therefore, the inequality equivalent to ∣ x − 1∣ < 5 is − 5 < x − 1 < 5 .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, if you are manufacturing parts for a machine, you might have a tolerance for the size of the parts. If the specification is that a part should be 10 cm long, but it can be off by up to 0.1 cm, then the actual length x must satisfy the inequality ∣ x − 10∣ < 0.1 . This ensures that the part will fit within the machine's specifications. Another example is in finance. Suppose you want to invest in a stock, but you want to limit your potential losses. You might set a rule that you will sell the stock if its price drops by more than 5% from your purchase price. If p is the purchase price and x is the current price, then you would sell if 0.05p"> ∣ x − p ∣ > 0.05 p .
