Question

problem 6 the grocery store sells one pound packages of ground beef. in order to be placed in their one pound display, the weight of ground beef packages must be within 0.15 pounds of the one pound target weight.
a. write an absolute value equation representing the expected upper and lower bounds.
b. solve the equation using a graphing calculator.
c. explain the meaning of the solutions from part b in terms of the applied context.
problems 7-10 solve the inequalities. represent the solution set on a number line.

1. ( 1|x + 5| > 2 )
2. ( 3|x| - 4 leq 2 )
3. ( 2|x - 4| < 8 )
4. ( -2|x| < 7 )

Explanation:

Problem 6

Part A

Step1: Define variables

Let \( x \) be the weight of the ground beef package. The target weight is \( 1 \) pound, and the difference between the package weight and the target weight must be within \( 0.15 \) pounds.

Step2: Write absolute value equation

The absolute value of the difference between \( x \) and \( 1 \) should be less than or equal to \( 0.15 \) (since it's within \( 0.15 \) pounds). So the equation is \( |x - 1| = 0.15 \) (for the bounds, we can also think of the inequality \( |x - 1| \leq 0.15 \), but the question says "equation representing the expected upper and lower bounds", so the equation for the bounds (the maximum and minimum) is \( |x - 1| = 0.15 \)).

Step1: Recall absolute value equation solution

For \( |a| = b \) (where \( b\geq0 \)), \( a = b \) or \( a = -b \). So for \( |x - 1| = 0.15 \), we have two cases.

Step2: Solve for \( x \) in each case

Case 1: \( x - 1 = 0.15 \)
\( x = 0.15 + 1 = 1.15 \)
Case 2: \( x - 1 = -0.15 \)
\( x = -0.15 + 1 = 0.85 \)
(Using a graphing calculator, we would graph \( y = |x - 1| \) and \( y = 0.15 \) and find their intersection points, which are at \( x = 0.85 \) and \( x = 1.15 \))

The solutions \( x = 0.85 \) and \( x = 1.15 \) represent the minimum and maximum weights (in pounds) that a ground beef package can have to be placed in the one - pound display. A package with a weight of \( 0.85 \) pounds is the lightest (lower bound) and a package with a weight of \( 1.15 \) pounds is the heaviest (upper bound) that still meets the store's requirement of being within \( 0.15 \) pounds of the one - pound target weight.

Answer:

\( |x - 1| = 0.15 \)

Part B