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For a large Thanksgiving dinner , Mr. G...

For a large Thanksgiving dinner , Mr. Gonzalez will cook a turkey that weighs at least 12 pounds but no more than 20 pounds. If x represents the weight of a turkey , in pounds, which of the following represents all possible values of x ?

A

|x-16| `ge ` 4

B

`|x-12| le 20`

C

`|x-16| le 4`

D

`|x-20| le 16`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine the range of possible weights for the turkey, represented by \( x \). The problem states that the turkey weighs at least 12 pounds and no more than 20 pounds. ### Step-by-Step Solution: 1. **Identify the lower limit**: - The turkey weighs at least 12 pounds. This can be expressed mathematically as: \[ x \geq 12 \] 2. **Identify the upper limit**: - The turkey weighs no more than 20 pounds. This can be expressed mathematically as: \[ x \leq 20 \] 3. **Combine the inequalities**: - We can combine the two inequalities to express the range of possible values for \( x \): \[ 12 \leq x \leq 20 \] 4. **Express the range in interval notation**: - The range of \( x \) can also be represented in interval notation as: \[ [12, 20] \] 5. **Express the range using absolute value**: - To express this range using absolute values, we can rewrite the inequalities. We can shift the interval to center around 16 (the midpoint of 12 and 20): \[ x - 16 \text{ must be between } -4 \text{ and } 4 \] - This can be expressed as: \[ -4 \leq x - 16 \leq 4 \] - This is equivalent to: \[ |x - 16| \leq 4 \] ### Final Answer: Thus, the representation of all possible values of \( x \) is: \[ |x - 16| \leq 4 \]
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