Question

a machine fills small bottles of glitter. the machine is set to fill each bottle with 3.25 oz of glitter, but it is acceptable to fill each bottle with 0.25 oz more or less than that. (a) let x be the amount of glitter (in ounces) in a bottle. write an absolute - value equation that can be used to find the minimum and maximum acceptable amount of glitter in a bottle. first choose an appropriate form. then fill in the blanks. |x - 3.25| = 0.25 (b) solve the equation from part (a) and graph the solution on the number line below.

Explanation:

Step1: Recall absolute - value equation concept

The general form of an absolute - value equation for the difference between a variable $x$ and a given value $a$ with a tolerance $b$ is $|x - a|=b$. Here, $a = 3.25$ (the target amount of glitter) and $b = 0.25$ (the acceptable deviation), so the equation $|x - 3.25|=0.25$ is correct.

Step2: Solve the absolute - value equation

If $|x - 3.25|=0.25$, then we have two cases:
Case 1: $x−3.25 = 0.25$. Solving for $x$, we add 3.25 to both sides of the equation: $x=0.25 + 3.25=3.5$.
Case 2: $x−3.25=-0.25$. Solving for $x$, we add 3.25 to both sides of the equation: $x=-0.25 + 3.25 = 3$.

Step3: Graph the solution on the number line

We mark the points $x = 3$ and $x = 3.5$ on the number line.

Answer:

The minimum acceptable amount of glitter is $3$ oz and the maximum acceptable amount is $3.5$ oz. On the number line, we mark the points $3$ and $3.5$ with closed - circles (since the values are included in the solution set).