Verified Answer :
- Define the "full box" condition as \(|w - 18| \le 0.25\).
- Determine the non-sellable condition by taking the negation, resulting in the inequality \(|w - 18| > 0.25\).
- Solve the inequality to find the range of weights: \(w < 17.75\) or \(w > 18.25\).
- Represent this on a number line with open circles at \(17.75\) and \(18.25\) and arrows pointing away from the center.
- The final range for non-sellable boxes is \(\boxed{w < 17.75 \text{ or } w > 18.25}\).
Explanation
- Analyzing the Weight Conditions
We are given that a cereal box is supposed to contain 18 ounces of cereal, but the actual weight can vary by up to \(\frac{1}{4}\) ounce while still being considered a "full box." This means the acceptable range for a full box is within \(\frac{1}{4}\) of 18 ounces. Let \(w\) be the weight of the cereal in ounces. The box is considered a full box if the distance between the actual weight \(w\) and the target weight 18 is less than or equal to \(\frac{1}{4}\). Mathematically, this is expressed as:
\[|w - 18| \le \frac{1}{4}\]
We want to identify the weights \(w\) of boxes that cannot be sold as full boxes, which means the condition for being a full box is not met. Therefore, we look for values of \(w\) that satisfy:
\[|w - 18| > \frac{1}{4}\]
- Solving the Inequality
To solve the inequality \(|w - 18| > \frac{1}{4}\), we must consider the definition of an absolute value inequality of the form \(|x| > a\), which is equivalent to \(x > a\) or \(x < -a\). In our case, \(x = w - 18\) and \(a = \frac{1}{4} = 0.25\). Thus, the inequality becomes:
\[w - 18 > 0.25 \quad \text{or} \quad w - 18 < -0.25\]
Solving these two individual inequalities:
- For \(w - 18 > 0.25\), adding 18 to both sides gives \(w > 18.25\).
- For \(w - 18 < -0.25\), adding 18 to both sides gives \(w < 17.75\).
So, the weights of boxes that are overfilled or underfilled are those in the set \(\{w \mid w < 17.75 \text{ or } w > 18.25\}\).
- Graphing the Solution
To represent the range of weights \(w < 17.75\) or \(w > 18.25\) on a graph, we use a number line.
- Since the inequality is strictly "greater than" or "less than" (not "or equal to"), we place open circles at \(17.75\) and \(18.25\) to indicate these specific points are excluded.
- For \(w < 17.75\), we draw a line segment extending to the left from the open circle at \(17.75\).
- For \(w > 18.25\), we draw a line segment extending to the right from the open circle at \(18.25\).
This graph clearly shows all the values that are either too light (underfilled) or too heavy (overfilled) to be sold as a full box. Thus, the condition for a non-sellable weight is \(\boxed{w < 17.75 \text{ or } w > 18.25}\).
Examples
This type of calculation is common in quality control within manufacturing processes. For instance, if a company produces batteries with a nominal voltage of 1.5 volts but allows a tolerance of 0.1 volts, any battery with a voltage outside the range \([1.4, 1.6]\) is considered defective. The inequality \(|V - 1.5| > 0.1\) helps engineers immediately identify which products need to be discarded. Understanding how to model these tolerances with absolute value inequalities ensures that production standards are met consistently and reliably.