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 0.5 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. \( \square \) - \( \square \) \( \leq \) \( \square \) \( \square \) \( + \) \( \square \) \( > \) \( \square \) \( \square \) - \( \square \) \( \square \) \( \square \) \( + \) \( \square \) \( \leq \) \( \square \) (b) Solve the absolute value inequality from part (a). \( \square \) and \( \square \) \( \square \) < \( \square>\square \) \( \square \) or \( \square \) \( \leq \square \) \( \geq \square \)

**Problem (a):** To determine the unacceptable amounts of soup in a can, we need to identify values of \( x \) (the amount of soup in ounces) that fall **outside** the acceptable range. The acceptable range is from \( 6 - 0.5 = 5.5 \) ounces to \( 6 + 0.5 = 6.5 \) ounces. An absolute value inequality captures the distance of \( x \) from the target value \( 6 \) ounces. Specifically, for unacceptable amounts, this distance should be **greater than** \( 0.5 \) ounces. The absolute value inequality representing the unacceptable amounts is: \[ |\, x - 6\,| \, > \, 0.5 \] --- **Problem (b):** To solve the inequality \( |\, x - 6\,| > 0.5 \), we consider the definition of absolute value inequalities: \[ |\, x - a\,| > b \quad \text{is equivalent to} \quad x - a > b \quad \text{or} \quad x - a < -b \] Applying this to our inequality: \[ |\, x - 6\,| > 0.5 \quad \Rightarrow \quad x - 6 > 0.5 \quad \text{or} \quad x - 6 < -0.5 \] Solving each part: 1. \( x - 6 > 0.5 \) \[ x > 6.5 \] 2. \( x - 6 < -0.5 \) \[ x < 5.5 \] **Final Solution:** \[ x < 5.5 \quad \text{or} \quad x > 6.5 \] This means that any can containing **less than 5.5 ounces** or **more than 6.5 ounces** of soup will fail inspection and be sent back. **Summary of Answers:** **(a)** \[ |\, x - 6\,| \, > \, 0.5 \] **(b)** \[ x < 5.5 \quad \text{or} \quad x > 6.5 \]