Acompany did a quality check on all the packs of nuts it manufactured. each pack of nuts is targeted to weigh 18.25 oz. a pack must weigh within 0.36 oz of the target weight to be accepted. what is the range of rejected masses, x, for the manufactured nuts? x < 17.89 or x > 18.61 because |x − 18.25| > 0.36 x < 17.89 or x > 18.61 because |x − 0.36| + 18.25 > 0 x < 18.25 or x > 18.61 because |x − 18.25| > 0.36 x < 18.25 or x > 18.61 because |x − 0.36| + 18.25 > 0
the first choice, |x − 18.25| > 0.36; x < 17.89 or x > 18.61
1. Target weight: 18.25 oz
2. Variability accepted: 0.36 oz
3. Range of accepted masses: 18.25 oz - 0.36 oz ≤ x ≤ 18.25 oz + 0.36 oz
4. Addition property of the inequalities (subtract 18.25 oz to the three parts of the inequality):
- 0.36 oz ≤ x - 18.25 oz ≤ 0.36 oz
5. Definition of absolute value inequality: | x - a | ≤ c equals - c ≤ x - a ≤ c
∴ - 0.36 ≤ x - 18.25 ≤ 0.36 equals | x - 18.25 | ≤ 0.36, which is the range of accepted masses.
6. The range of rejected masses is the complement, so it is:
|x - 18.25 | > 0.36
7. Solve the inequality to find the range of rejected masses:
a) x - 18.25 > 0.36 ⇒ x > 18.25 + 0.36 ⇒ x > 18.61
b) x - 18.25 < 0.36 ⇒ x < 18.25 - 0.36 ⇒ x < 17.89
or , since
According to the question the weight is rejected if it is more than 0.36 oz, the weight should not be beyond 0.36 oz that is from the target weight of 18.25.
We can say that, the difference between the actual weight and the target weight is greater than 0.36 oz.
Lets say the actual weight is 'x'. So,
But there can be times when the actual weight 'x' could be less than 0.36, So the order will be changed:
The absolute value will sum up these two into a single inequality:
Simplifying the inequalities, we get:
(we have added 18.25 on both sides)
Subtracting 18.25 from both the sides we get:
After sign change it is:
So we can see that only or since is the one that represents the scenario given above.
choice b is correct
we have been given the following system of linear inequalities;
we are required to graphically determine the solution to these system of inequalities. using a graphing tool, the solution is as shown in the attachment below;