Jack is designing the manufacturing that will produce packing material for a particular computer. The packing material needs to be within 0.5 mm of the desired thickness, which is 27.5 mm.Which equation can Jack use to represent the limits of the width, w, of the packing material?A.|w+27.5|=0.5B.|w−0.5|=27.5C.|w+0.5|=27.5D.|w−27.5|=0.5

The key to solving this problem is understanding that Jack wants the thickness of the packing material to be within **0.5 mm** of the desired thickness, which is **27.5 mm**. This is a classic absolute value problem, where the actual thickness \( w \) must be within 0.5 mm of 27.5 mm.

### Absolute value concept:
The absolute value equation for such problems is of the form:
\[
| w - \text{desired value} | = \text{tolerance}
\]
Where:
- \( w \) is the actual thickness.
- \( \text{desired value} = 27.5 \) mm (the ideal thickness).
- \( \text{tolerance} = 0.5 \) mm (the allowable deviation).

Thus, the equation that represents this situation is:
\[
| w - 27.5 | = 0.5
\]

This shows that the thickness \( w \) can be no more than 0.5 mm greater or less than 27.5 mm.

### Correct Answer:
The correct equation is **D. \( |w - 27.5| = 0.5 \)**.

Would you like more details or have any questions?

Here are five related questions:
1. What are the solutions to the equation \( |w - 27.5| = 0.5 \)?
2. How does the absolute value function help model real-world tolerances?
3. Can you graph the equation \( |w - 27.5| = 0.5 \) and explain what the graph represents?
4. What is the difference between equations that use absolute values and those that don’t?
5. How would you set up a similar equation if the tolerance were different, say 1 mm?

**Tip:** When solving absolute value equations, remember that \( |x| = a \) leads to two cases: \( x = a \) or \( x = -a \).

Jack is designing the manufacturing that will produce packing material for a particular computer. The packing material needs to be within 0.5 mm of the desired thickness, which is 27.5 mm. Which equation can Jack use to represent the limits of the width, w, of the packing material?
A.|w+27.5|=0.5
B.|w−0.5|=27.5
C.|w+0.5|=27.5
D.|w−27.5|=0.5

Understand how to solve the absolute value equation |w - 27.5| = 0.5, which models the tolerance limits for packing material thickness in a manufacturing process. Learn key concepts in algebra and absolute value theory, suitable for Grades 6-8 students.

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