a manufacturing process calls for the width of a part to be 10 cm. any part manufactured with an error of greater than 0.13 cm is rejected. parts with what widths are rejected?
write and solve an absolute value inequality for this situation
10 - .13 < w < 10 + .13
To determine which widths are rejected, we need to find the range of widths that exceed the allowed error of 0.13 cm.
Let's denote the width of a part as "x". Since the required width of the part is 10 cm, we can subtract the allowed error from the required width to find the lower limit of the accepted width:
10 - 0.13 = 9.87 cm
So, any part with a width less than 9.87 cm is rejected.
Now, let's find the upper limit of the accepted width by adding the allowed error to the required width:
10 + 0.13 = 10.13 cm
So, any part with a width greater than 10.13 cm is also rejected.
Therefore, the absolute value inequality for this situation is:
| x - 10 | > 0.13
This inequality states that the absolute difference between the width (x) and the required width (10) should be greater than 0.13 cm.
To find the range of widths that are rejected, we will use an absolute value inequality. Let's set up the absolute value inequality and solve for the rejected widths.
Let's denote "x" as the width of the part.
The given manufacturing process requires the width of the part to be 10 cm. So, the target width is 10 cm.
According to the condition, any part manufactured with an error of greater than 0.13 cm is rejected. This means that the acceptable range is within 0.13 cm of the target width, both above and below.
To represent this using an absolute value inequality, we can write:
| x - 10 | > 0.13
To solve this absolute value inequality, we can break it into two cases:
Case 1: x - 10 > 0.13
In this case, x is greater than the target width plus 0.13, which would mean the width is outside the acceptable range:
x - 10 > 0.13
x > 10 + 0.13
Case 2: - (x - 10) > 0.13
In this case, x is less than the target width minus 0.13, which would also mean the width is outside the acceptable range:
-(x - 10) > 0.13
-x + 10 > 0.13
-x > 0.13 - 10
-x > -9.87
To isolate x, both cases require multiplying by -1 and flipping the inequality sign:
Case 1: x > 10 + 0.13
x > 10.13
Case 2: x < -(-9.87)
x < 9.87
Thus, the rejected widths would be any widths greater than 10.13 cm or less than 9.87 cm.