Choose the compound inequality that can be used to solve the original inequality $|3x-5| > 10$.

A. $-10 < |3x-5| < 10$
B. $3x-5 > -10$ or $3x-5 < 10$
C. $3x-5 < -10$ or $3x-5 > 10$
D. $-10 < 3x-5 < 10$

The correct compound inequality for the absolute value inequality 10"> ∣3 x − 5∣ > 10 is 3 x − 5 < − 10 or 10"> 3 x − 5 > 10 . Thus, the answer is Option C. This reflects that 3 x − 5 must be either less than − 10 or greater than 10 .
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Answered by Anonymous | 2025-07-04

The problem involves solving an absolute value inequality.
The absolute value inequality 10"> ∣3 x − 5∣ > 10 is equivalent to 10"> 3 x − 5 > 10 or 3 x − 5 < − 10 .
Therefore, the correct compound inequality is 3 x − 5 < − 10 or 10"> 3 x − 5 > 10 .
The final answer is 10}"> 3 x − 5 < − 10 or 3 x − 5 > 10 ​ .

Explanation

Understanding the Problem We are given the absolute value inequality 10"> ∣3 x − 5∣ > 10 and need to find the equivalent compound inequality.

Applying the Absolute Value Rule The absolute value inequality a"> ∣ x ∣ > a is equivalent to a"> x > a or x < − a . Applying this rule to our inequality, we get 10"> 3 x − 5 > 10 or 3 x − 5 < − 10 .

The Solution Therefore, the correct compound inequality is 3 x − 5 < − 10 or 10"> 3 x − 5 > 10 .


Examples
Absolute value inequalities are useful in many real-world scenarios. For example, a machine that produces bolts needs to make bolts that are very close to a certain diameter. If the diameter is too far from the specified value, the bolts are unusable. We can express this situation using an absolute value inequality, where the absolute value represents the difference between the actual diameter and the specified diameter. If we want the difference to be less than a certain tolerance, we can set up an inequality like ∣ d − s ∣ < t , where d is the actual diameter, s is the specified diameter, and t is the tolerance. Solving this inequality will give us the range of acceptable diameters for the bolts.

Answered by GinnyAnswer | 2025-07-04