We have to replace a valve in a space shuttle the broken valve had a diameter of 6 mm the diameter of a new valve for the space shuttle can be slightly different size but must be within 0.001 mm of 6 mm in order for it to work Right and solve an absolute value equation with the variable inside the absolute value to find the boundary values for the acceptable diameters of the valve

The problem requires us to find acceptable diameters for the new valve, which must be within 0.001 mm of 6 mm. This is an absolute value problem where we need to find the boundary values for the acceptable range.

Let's define the variable \( x \) as the diameter of the new valve. The diameter must satisfy the condition that it is within 0.001 mm of 6 mm. This can be written as:

\[
|x - 6| \leq 0.001
\]

### Step 1: Solve the absolute value inequality
To solve \( |x - 6| \leq 0.001 \), we break it into two inequalities:

\[
-0.001 \leq x - 6 \leq 0.001
\]

### Step 2: Solve for \( x \)
Now, solve each part separately:

- For the left side: 
  \[
  -0.001 \leq x - 6 \implies x \geq 6 - 0.001 \implies x \geq 5.999
  \]

- For the right side:
  \[
  x - 6 \leq 0.001 \implies x \leq 6 + 0.001 \implies x \leq 6.001
  \]

### Final solution:
The acceptable diameters for the new valve are between 5.999 mm and 6.001 mm. Therefore, the boundary values for the diameter are:

\[
5.999 \leq x \leq 6.001
\]

This is the range within which the new valve's diameter must lie for it to work properly.

Would you like more details or have any questions?

### Related questions:
1. How do you solve absolute value inequalities in general?
2. What is the difference between absolute value equations and inequalities?
3. Can this method be used for tolerances in other engineering problems?
4. How would the problem change if the tolerance was 0.002 mm instead?
5. What happens if the valve's diameter exceeds the tolerance limit?

### Tip:
When solving absolute value inequalities, always remember to break the equation into two parts (positive and negative) to explore both possible ranges of values.

We have to replace a valve in a space shuttle. The broken valve had a diameter of 6 mm. The diameter of a new valve for the space shuttle can be slightly different in size but must be within 0.001 mm of 6 mm in order for it to work. Write and solve an absolute value equation with the variable inside the absolute value to find the boundary values for the acceptable diameters of the valve.

Solve an absolute value inequality to find acceptable diameters for a space shuttle valve. Learn how to calculate the boundary values within 0.001 mm of 6 mm using algebraic concepts and absolute value properties. Suitable for students in Grades 9-10.

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