On any given day, the number of users, [tex]$u$[/tex], that access a certain website can be represented by the inequality [tex]$|125-u| \leq 30$[/tex]. Which of the following represents the range of users that access the website each day?
A. [tex]$u \geq 95$[/tex] or [tex]$u \leq 155$[/tex]
B. [tex]$95 \geq u \geq 155$[/tex]
C. [tex]$u \leq 95$[/tex] or [tex]$u \geq 155$[/tex]
D. [tex]$95 \leq u \leq 155$[/tex]
Answer (2)
Rewrite the absolute value inequality as a compound inequality: − 30 ≤ 125 − u ≤ 30 .
Isolate u by subtracting 125 from all parts: − 155 ≤ − u ≤ − 95 .
Multiply by -1 to solve for u , reversing the inequality signs: 155 g e ug e 95 .
Rewrite the inequality: 95 ≤ u ≤ 155 . The range of users is 95 ≤ u ≤ 155 .
Explanation
Understanding the Problem We are given the inequality ∣125 − u ∣ ≤ 30 and need to find the range of values for u . This involves solving an absolute value inequality.
Rewriting the Inequality The absolute value inequality ∣125 − u ∣ ≤ 30 can be rewritten as a compound inequality: − 30 ≤ 125 − u ≤ 30
Isolating u To isolate u , we first subtract 125 from all parts of the inequality: − 30 − 125 ≤ 125 − u − 125 ≤ 30 − 125 − 155 ≤ − u ≤ − 95
Multiplying by -1 Now, we multiply all parts of the inequality by -1. Remember that multiplying by a negative number reverses the direction of the inequality signs: ( − 1 ) ( − 155 ) g e ( − 1 ) ( − u ) g e ( − 1 ) ( − 95 ) 155 g e ug e 95
Final Solution We can rewrite this inequality as: 95 ≤ u ≤ 155 This means that u is greater than or equal to 95 and less than or equal to 155.
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, if a machine is supposed to produce bolts that are 5 cm long, but the acceptable tolerance is ± 0.2 cm, then the length L of the bolts must satisfy the inequality ∣ L − 5∣ ≤ 0.2 . This ensures that the bolts are within the acceptable range of lengths. Similarly, in manufacturing or quality control, absolute value inequalities help define acceptable ranges for measurements.
To solve the inequality ∣125 − u ∣ ≤ 30 , we rewrote it as a compound inequality and found that the range of users u is 95 ≤ u ≤ 155 . This means the number of users accessing the site daily is between 95 and 155 inclusive. Thus, the correct answer is option D.
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