Which compound inequality can be used to solve the inequality $|3 x+2|>7$?
A. $-7<3 x+2>7$
B. $-7>3 x+2>7$
C. $3 x+2>-7$ or $3 x+2>7$
D. $3 x+2<-7$ or $3 x+2>7$
Answer (2)
The absolute value inequality 7"> ∣3 x + 2∣ > 7 is given.
Recall that a"> ∣ u ∣ > a is equivalent to u < − a or a"> u > a .
Apply this to get 3 x + 2 < − 7 or 7"> 3 x + 2 > 7 .
Therefore, the correct compound inequality is 3 x + 2 < − 7 or 7"> 3 x + 2 > 7 .
Explanation
Understanding the Problem We are given the absolute value inequality 7"> ∣3 x + 2∣ > 7 and asked to find the equivalent compound inequality.
Absolute Value Inequality Recall that for any expression u and any positive number a , the absolute value inequality a"> ∣ u ∣ > a is equivalent to the compound inequality u < − a or a"> u > a .
Applying the Definition Applying this to our problem, where u = 3 x + 2 and a = 7 , we have 3 x + 2 < − 7 or 7"> 3 x + 2 > 7 .
Examples
Absolute value inequalities are useful in many real-world situations. For example, suppose a machine is designed to fill bags with 500 grams of sugar, but the actual weight can vary by up to 5 grams. This means the difference between the actual weight w and the target weight 500 must be less than or equal to 5, which can be written as ∣ w − 500∣ ≤ 5 . Solving this inequality helps determine the range of acceptable weights for the bags of sugar.
The compound inequality for the absolute value inequality 7"> ∣3 x + 2∣ > 7 is 3 x + 2 < − 7 or 7"> 3 x + 2 > 7 . Therefore, the correct answer is option D. This captures all necessary values of x .
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