The diameter of a valve for the space shuttle must be within 0.001 mm of 5 mm. Write and solve an absolute-value equation to find the boundary values for the acceptable diameters of the valve.
|d-5.000| <= 0.001
So,
d - 5 = 0.001
d * 1000 - 5 * 1000 = 0.001 * 1000
1000d - 5000 = 1
1000d - 5000 + 5000 = 1 + 5000
1000d = 5001
1000d / 1000 = 5001 / 1000
d = 5001/1000
or
d - 5 = -0.001
d * 1000 - 5 * 1000 = -0.001 * 1000
1000d - 5000 = -1
1000d - 5000 + 5000 = -1 + 5000
1000d = 4999
100d / 1000 = 4999 / 1000
d = 4999 / 1000
Would this be the correct solution?
not quite. You want d to be between those values:
4.999 <= d <= 5.001
To find the boundary values for the acceptable diameters of the valve, we can write an absolute value equation based on the given requirement.
Let's assume the diameter of the valve is "x" mm. According to the given requirement, the diameter must be within 0.001 mm of 5 mm.
To find the acceptable boundary values, we can write the absolute value equation:
| x - 5 | ≤ 0.001
To solve this equation, we consider two cases:
Case 1: (x - 5) ≥ 0
In this case, the absolute value equation becomes:
x - 5 ≤ 0.001
To solve for x, we isolate the variable by adding 5 to both sides:
x ≤ 5 + 0.001
x ≤ 5.001
Therefore, in this case, the acceptable diameter has an upper boundary of 5.001 mm.
Case 2: (x - 5) < 0
In this case, the absolute value equation becomes:
- (x - 5) ≤ 0.001
Simplifying the inequality:
- x + 5 ≤ 0.001
To solve for x, we isolate the variable by subtracting 5 from both sides:
- x ≤ 0.001 - 5
- x ≤ -4.999
When we multiply both sides by -1, the inequality sign flips:
x ≥ 4.999
Therefore, in this case, the acceptable diameter has a lower boundary of 4.999 mm.
In summary, the absolute-value equation | x - 5 | ≤ 0.001 gives us the acceptable boundary values for the diameter of the valve: 4.999 mm ≤ x ≤ 5.001 mm.