Problem 46 - Algebra 2

Question

A machine fills bags with about 16 ounces of sugar each. After the bags are filled, another machine weighs them. If the bag weighs 0.3 ounce more or less than the desired weight, the bag is rejected. Write an equation to find the heaviest and lightest bags the machine will approve.

Step-by-Step Solution

Answer

The heaviest is 16.3 oz, and the lightest is 15.7 oz.

Step 1: Understanding the Problem

We're asked to find the range of weights that a machine considers acceptable for bags of sugar. The machine fills the bags with 16 ounces of sugar, and a bag is approved if it weighs no more than 0.3 ounces more or less than 16 ounces.

Step 2: Setting Up the Equation

Since the weight of the bag needs to be within 0.3 ounces of 16 ounces, we can express the acceptance criteria as an inequality: \( |w - 16| \leq 0.3 \) where \( w \) is the weight of the bag.

Step 3: Solving the Inequality

We solve \( |w - 16| \leq 0.3 \) by splitting it into two inequalities: \( w - 16 \leq 0.3 \) and \( w - 16 \geq -0.3 \).

Step 4: Solving the First Inequality

The inequality \( w - 16 \leq 0.3 \) simplifies to \( w \leq 16.3 \).

Step 5: Solving the Second Inequality

The inequality \( w - 16 \geq -0.3 \) simplifies to \( w \geq 15.7 \).

Step 6: Combining the Results

Combining the results of the two inequalities, we find that the acceptable weight range is \( 15.7 \leq w \leq 16.3 \).

Key Concepts

Absolute Value

When dealing with absolute value, it's important to understand that it represents the distance of a number from zero on a number line. In a mathematical sense, the absolute value of a number is always positive. So, if you see something like \(|w - 16|\), it describing how far "w" is from the number 16, regardless of whether "w" is larger or smaller.

• Absolute value can be thought of as magnitude without direction.
• It’s particularly useful in problems involving tolerances or ranges, like the weight of a bag.

When the problem states \(|w - 16| \leq 0.3\), this means the weight "w" is allowed to be 0.3 units (or ounces, in this case) above or below 16 ounces. Absolute values are handy in defining these ranges because they inherently consider both over and under variations.

Inequality Solving

Solving inequalities involves finding a set of values that satisfy an inequality equation. When you have an absolute value inequality like \(|w - 16| \leq 0.3\), you're actually dealing with two separate inequalities. This is because the absolute value encompasses both positive and negative deviations from a point.

• First, split the absolute inequality: \(w - 16 \leq 0.3\) and \(w - 16 \geq -0.3\).
• Then solve each inequality separately just like a regular linear equation.
• Keep the inequality signs in mind; treating them as equalities can lead to incorrect solutions.

In our sugar bag scenario, we're solving for the range of acceptable weights. By solving \(w - 16 \leq 0.3\), we determine that the maximum weight is 16.3 ounces.
On the flip side, solving \(w - 16 \geq -0.3\) gives us the minimum weight, 15.7 ounces. These solutions together provide the full range of acceptable weights.

Range of Values

Finding the range of values is a common task in mathematics, especially when assessing conditions that involve limits or tolerances. Once you've solved the inequalities, you combine the solutions to find this range.

• This tells us all the feasible values that meet our criteria.
• In this context, it represents all the weights that are acceptable for the bags.

For the sugar bags example, after solving the inequalities, we conclude that bags weighing between 15.7 and 16.3 ounces are accepted.
This range captures all possible approved weights, showing not only the central target weight but also the allowable deviation. It provides a clear guideline for ensuring each bag's weight is within specifications.