Question

directions: read each question carefully. use desmos as much as possible to help you answer each question.
1.
a company makes and boxes spaghetti.

• one machine fills each box with approximately 32 ounces of spaghetti.
• after the boxes are filled, another machine weighs each box.
• a box is discarded if the weight of the box differs by more than 0.25 ounce from the target weight of 32 ounces.

which inequality can be used to find the range of acceptable weights, x, of the spaghetti?
a |x - 0.25| ≤ 32
b |x + 0.25| ≤ 32
c |x - 32| ≤ 0.25
d |x + 32| ≤ 0.25
2.
the graph of an equation is shown below.
graph of a v - shaped graph (absolute value graph) with grid
which equation represents the graph?
a y = |x| - 2
b y = |2x| - 2
c y = |x - 2|
d y = |2x - 2|
3.
a company makes and sells candles. each candle should weigh 290 grams, but an error of 4 grams is allowed. which inequality can be used to determine the acceptable weight, x?
a |x - 290| ≤ 4
b |290 - x| ≥ 4
c |x - 4| ≤ 290
d |4 - x| ≥ 290

Explanation:

Question 1

Step1: Understand absolute value inequality for error

The absolute value inequality \(|x - \text{target}| \leq \text{error}\) represents the acceptable range where \(x\) is the measured value, target is the target value, and error is the allowed difference. Here, target weight is 32 ounces, error is 0.25 ounce. So the inequality should be \(|x - 32| \leq 0.25\), which is option C.

Step1: Analyze the vertex and slope of the absolute - value graph

The parent function of absolute value is \(y = |x|\), with vertex at \((0,0)\) and slope \(\pm1\). The given graph has vertex at \((0, - 2)\) and the slope of the lines (for \(x>0\) and \(x < 0\)) is 2 (since from \(x = 0\) to \(x=1\), \(y\) changes from - 2 to 0, so slope \(m=\frac{0 - (-2)}{1-0}=2\)). The general form of an absolute - value function is \(y=a|x - h|+k\), where \((h,k)\) is the vertex. Here \(h = 0\), \(k=-2\) and \(a = 2\). So the function is \(y = |2x|-2\) (because \(y=a|x - h|+k\) with \(a = 2\), \(h = 0\), \(k=-2\) gives \(y=|2x|-2\)), which is option B.

Step1: Apply absolute - value inequality for error

The formula for the acceptable range of a measured value \(x\) with target value \(t\) and error \(e\) is \(|x - t|\leq e\). Here, the target weight \(t = 290\) grams and the error \(e = 4\) grams. So the inequality is \(|x - 290|\leq4\), which is option A.

Answer:

C. \(|x - 32| \leq 0.25\)

Question 2