Question

A company weighs each 80 -ounce bag of sugar it produces. After production, any bag that does not weigh within \(1.2\) ounces of 80 ounces cannot be sold. Solve the equation \(|x-80|=1.2\) to find the least and greatest acceptable weights of an 80 -ounce bag of sugar.

Short answer

The least and greatest acceptable weights of an 80 -ounce bag of sugar are 78.8 ounces and 81.2 ounces respectively.

Step-by-step solution

Understand the equation

The equation \(|x-80|=1.2\) is equivalent to either \(x-80=1.2\) or \(x-80=-1.2\), because the absolute value function measures the distance between two numbers on a number line, and this distance can be either positive or negative.

Solve for the first acceptable weight

Solving the equation \(x-80=1.2\), we isolate \(x\) by adding 80 to both sides, resulting in \(x = 80 + 1.2 = 81.2\). This indicates that the greatest acceptable weight for the bag of sugar is 81.2 ounces.

Solve for the second acceptable weight

Solving the equation \(x-80=-1.2\), we isolate \(x\) by adding 80 to both sides, resulting in \(x = 80 - 1.2 = 78.8\). This indicates that the least acceptable weight for the bag of sugar is 78.8 ounces.

Key concepts

Solving Absolute Value Equations

When dealing with absolute value equations, such as in the weight tolerance problem presented by a company's quality control, it's important to understand the basic definition. The absolute value of a number represents its distance from zero on a number line, regardless of direction.

In the equation \( |x - 80| = 1.2 \), the absolute value indicates all the numbers that are exactly 1.2 units away from 80. To solve this, we consider both the positive and negative scenarios. This leads to two separate equations: \( x - 80 = 1.2 \) and \( x - 80 = -1.2 \). By solving these, we find that the acceptable weights for the bag of sugar are 81.2 ounces and 78.8 ounces.

This approach allows us to visualize the problem on a number line, where we'd find two points representing the acceptable range for the bags of sugar. It's crucial to interpret the absolute value as a distance when solving these types of questions.

Algebraic Reasoning

Algebraic reasoning requires logical thinking to solve equations and understand the relationships between different elements within a problem. For our example with the weight of sugar bags, algebraic reasoning guided us through the process of evaluating the meaning behind the absolute value and translating it into two possible situations that a bag's weight can fall into.

It's the act of breaking down the equation into simpler parts that can be individually solved. Here, we visualized the equation in terms of 'if' scenarios – if the weight is above the standard, or if it is below. Moreover, algebraic reasoning helps ensure that all possible solutions are considered. In other words, we explore the realm of 'what could be' based on the given information and mathematical rules, leading to a comprehensive solution.

Encouraging this type of reasoning is vital, as it not only assists students in solving algebraic problems but also in applying logical strategies to real-world situations.

Weight Tolerance Problem-Solving

In industrial settings and quality control, weight tolerance problem-solving is critical, as seen in our sugar bag example. Tolerance defines the acceptable range of deviation from a nominal value. In this case, the nominal weight is 80 ounces, with a tolerance of ±1.2 ounces.

To solve this, we convert the practical problem into an algebraic equation where the absolute value represents the tolerance limits. Through our solutions, we discovered that the bags must weigh between 78.8 and 81.2 ounces to be considered within tolerance.

This application of mathematics is not limited to sugar bags; it's a concept used in various aspects of manufacturing and engineering to maintain quality and consistency. The ability to solve these algebraic equations directly translates to ensuring products meet specific standards, which is indispensable in many industries.