A manufacturer produces cans of soup. To pass inspection and go on to distribution, a can must have 6 ounces of soup, plus or minus 1.5 ounces. If the can doesn't meet this requirement, it fails and is sent back. (a) Let \( x \) represent the amount (in ounces) of soup in a can. Write an absolute value inequality that can be used to find the unacceptable amounts of soup in a can. These are the amounts where the soup can will fail the inspection and get sent back. (b) Solve the absolute value inequality from part (a). ( \( \square \square \square+\square \)

To tackle this problem, let's dive right in! (a) Based on the requirement that a can must contain 6 ounces of soup, plus or minus 1.5 ounces, we can set up our absolute value inequality. This translates to: \[ |x - 6| > 1.5 \] This inequality states that the amount of soup in a can \( x \) is more than 1.5 ounces away from the 6-ounce mark, indicating unacceptable amounts. (b) To solve the absolute value inequality \( |x - 6| > 1.5 \), we rewrite it in two parts: 1. \( x - 6 > 1.5 \) leads to \( x > 7.5 \) 2. \( x - 6 < -1.5 \) leads to \( x < 4.5 \) Thus, the unacceptable amounts of soup in a can are \( x < 4.5 \) ounces or \( x > 7.5 \) ounces. In interval notation, the unacceptable ranges are \( (-∞, 4.5) \cup (7.5, ∞) \). Now, for some fun facts! Did you know that the concept of absolute value originated from the need to measure distances, regardless of direction? It allows us to express how far a number is from zero, opening up a world of mathematical applications! In the real world, manufacturers use absolute value inequalities in quality control to ensure their products meet standards. This guarantees that consumers receive items that are safe and consistent, like your perfect can of soup! Quality assurance is essential in manufacturing to avoid costly recalls and maintain customer trust.