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 $$ or $$ \square $$ $$ \leq \square $$ $$ \geq \square $$

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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 $$ \square $$ $$ \leq \square $$ $$ \geq \square $$

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**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)** $$ x < 5.5 \quad \text{or} \quad x > 6.5 $$ **(a)** $$ |\, x - 6\,| \, > \, 0.5 $$ **(b)** $$ x < 5.5 \quad \text{or} \quad x > 6.5 $$

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