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A company manufactures cell phone cases. the length of a certain case must be with in  of  as shown. All cases with lengths outside of this range are removed from the inventory How could you use an absolute value inequality to represent the lengths of all the cases that should be removed ? 


To solve the problem, we need to find an absolute value inequality that represents the lengths of the cell phone cases that should be removed.

### Problem Breakdown:
- **Target Length:** 125 mm
- **Tolerance:** ±0.25 mm

This means that the acceptable range for the lengths of the cell phone cases is from \( 125 - 0.25 = 124.75 \) mm to \( 125 + 0.25 = 125.25 \) mm.

### Absolute Value Inequality:
An absolute value inequality that represents the acceptable lengths \( x \) of the cases is:
\[
|x - 125| \leq 0.25
\]
This inequality states that the difference between any case's length \( x \) and the target length of 125 mm must be within 0.25 mm.

### Cases to be Removed:
The question asks for the lengths of all the cases that should be removed, which corresponds to those lengths that fall outside the acceptable range. To express this, we need the inequality for cases that do **not** meet the requirement:
\[
|x - 125| > 0.25
\]
This inequality represents all the lengths \( x \) that are either less than 124.75 mm or greater than 125.25 mm, meaning they fall outside the acceptable range and should be removed.

### Explanation:
- The absolute value \( |x - 125| \) measures how far the length \( x \) is from the target length of 125 mm.
- The inequality \( |x - 125| > 0.25 \) specifies that the length \( x \) is more than 0.25 mm away from 125 mm, either above or below, meaning it’s unacceptable and should be removed from the inventory.

Would you like any more details on this, or do you have any further questions?

### Related Questions:
1. How can you rewrite the absolute value inequality \( |x - 125| \leq 0.25 \) as two separate linear inequalities?
2. What would the acceptable range be if the tolerance was increased to ±0.5 mm?
3. How would the inequality change if the target length was 130 mm with the same tolerance?
4. Can you derive the acceptable length range if the tolerance was given as a percentage instead of a fixed value?
5. How does changing the tolerance affect the range of acceptable lengths?

### Tip:
When working with absolute value inequalities, always think about the expression as measuring a "distance" from a central value, which helps in understanding the acceptable range.

A company manufactures cell phone cases. The length of a certain case must be within 0.25 mm of 125 mm. All cases with lengths outside of this range are removed from the inventory. How could you use an absolute value inequality to represent the lengths of all the cases that should be removed? Explain.

Learn how to represent the lengths of cell phone cases that should be removed using an absolute value inequality. This problem involves understanding the concepts of absolute value, range, and inequality, and is suitable for students in Grades 7-9.

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