Let's break down the problem step by step.

Problem Overview:

A machine fills small bottles of glitter, aiming for 2 ounces of glitter per bottle, but it is acceptable to fill each bottle with $\pm 0.5$ ounces. We are tasked with:

1. (a) Writing an absolute value equation that represents this situation.
2. (b) Solving and graphing the equation from part (a).

Part (a) - Absolute Value Equation

Let $x$ represent the amount of glitter in ounces in a bottle. The machine is set to fill 2 oz, but it can vary by $\pm 0.5$ ounces.

An absolute value equation for this tolerance range can be written as: $|x - 2| = 0.5$ This equation says that the difference between the actual amount of glitter $x$ and the target amount 2 oz must be 0.5 oz or less.

Part (b) - Solve the Equation

Now, solve the equation $|x - 2| = 0.5$.

To solve this: $x - 2 = 0.5 \quad \text{or} \quad x - 2 = -0.5$ Solving each equation:

1. $x - 2 = 0.5$ implies $x = 2.5$
2. $x - 2 = -0.5$ implies $x = 1.5$

Thus, the amount of glitter can range between 1.5 oz and 2.5 oz.

Graphing the Solution

The solution is $1.5 \leq x \leq 2.5$. On a number line, this is represented by a line segment between 1.5 and 2.5, inclusive. The endpoints 1.5 and 2.5 should be marked with closed circles to indicate that these values are included in the solution.

Would you like further details, or have any other questions? Below are some related questions for deeper understanding:

1. How do we derive absolute value equations from real-world contexts?
2. How do we interpret the graph of an absolute value inequality on a number line?
3. What are other methods to solve absolute value equations?
4. How would the equation change if the acceptable variation was $\pm 0.3$ ounces?
5. What would the equation look like if the machine allowed a tolerance of $\pm 1$ ounce?

Tip: When solving absolute value equations, remember that the expression inside the absolute value can be equal to both the positive and negative values of the number on the other side of the equation!