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The problem asks how to use an absolute value inequality to represent the lengths of all the phone cases that should be removed from the inventory, based on a length requirement of 125 mm with a tolerance of ±0.25 mm.

### Step-by-Step Explanation:
1. **Identify the Range**: The acceptable range of lengths for the phone cases is from 125 mm - 0.25 mm to 125 mm + 0.25 mm, which is:
   - Lower limit: \(125 \, \text{mm} - 0.25 \, \text{mm} = 124.75 \, \text{mm}\)
   - Upper limit: \(125 \, \text{mm} + 0.25 \, \text{mm} = 125.25 \, \text{mm}\)

   So, the lengths of the cases should be within the range of 124.75 mm to 125.25 mm.

2. **Set up the Absolute Value Inequality**: 
   To express this as an absolute value inequality, let \(x\) represent the length of a case. The inequality for the lengths that fall within this range can be written as:
   \[
   |x - 125| \leq 0.25
   \]
   This inequality means that the distance between the length \(x\) and 125 mm should be no more than 0.25 mm.

3. **Determine Cases to Remove**: 
   The lengths that fall outside this range are the cases that should be removed. These lengths would satisfy the inequality:
   \[
   |x - 125| > 0.25
   \]
   This inequality represents the lengths of cases that are either less than 124.75 mm or greater than 125.25 mm.

### Final Answer:
The absolute value inequality that represents the lengths of all the cases that should be removed from the inventory is:
\[
|x - 125| > 0.25
\]
This inequality captures all the phone cases with lengths that are either too short or too long compared to the acceptable range.

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Do you have any questions or need more details on this? Here are five related questions to explore further:

1. What is the mathematical definition of absolute value, and how is it applied in inequalities?
2. How would the inequality change if the tolerance was reduced to ±0.1 mm?
3. What is the significance of interpreting the absolute value inequality in the context of real-world problems like quality control?
4. How can this concept be applied to other dimensions, such as width and height of the phone cases?
5. What are some potential challenges when dealing with absolute value inequalities in manufacturing?

**Tip**: When solving absolute value inequalities, remember that the inequality can be split into two separate linear inequalities, which can be solved individually for a clearer understanding.

A company manufactures cell phone cases. The length of a certain case must be within 0.25 mm of 125 mm, as shown in the figure. 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?

Learn how to use absolute value inequalities to determine which cell phone cases should be removed from inventory. This problem covers key algebra concepts like absolute value and inequalities, focusing on practical applications in quality control. Suitable for students in Grades 7-9.

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