To determine which statement is true regarding the number of candles (x) and ornaments (y) that Sylvia bought, we consider the constraints based on the information provided:
- Sylvia spent a total of $50 on candles and ornaments.
- The cost of candles is $5 each, and the cost of ornaments is $7 each.
- The total number of items (candles and ornaments) she bought is 8.
From this information, we can identify the following constraints:
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Total cost constraint: \[ 5x + 7y \leq 50 \]
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Total items constraint: \[ x + y = 8 \]
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Non-negativity constraint: \[ x \geq 0 \quad \text{and} \quad y \geq 0 \]
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Item upper limit constraint: Since she bought a total of 8 items, it follows that: \[ x \leq 8 \quad \text{and} \quad y \leq 8 \]
Considering the options provided:
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"The inequality that represents the constraint on the number of candles is \( x \leq 8 \) and on the number of ornaments is \( y \leq 8 \) because the candles and ornaments together cannot be more than 8 items." - True
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"The inequality that represents the constraint on the number of candles is \( x \) is less than or equal to 8 and on the number of ornaments is \( y \) is less than or equal to 8 because the candles and ornaments together cannot be more than 8 items." - Also True, but the format is slightly different.
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"The inequality that represents the constraint on the number of candles is \( x > 0 \) and on the number of ornaments is \( y > 0 \) because there must be a positive number of candles or ornaments." - False, as she could buy zero of either.
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"The inequality that represents the constraint on the number of candles is \( x \) is greater than 0 and on the number of ornaments is \( y \) is greater than 0 because there must be a positive number of candles or ornaments." - False, similar reasoning.
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"The inequality that represents the constraint on the number of candles is \( x \geq 0 \) and on the number of ornaments is \( y \geq 0 \) because there cannot be a negative number of candles or ornaments." - True.
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"The inequality that represents the constraint on number of candles is \( 0 \leq x \leq 8 \) and on the number of ornaments is \( 0 \leq y \leq 8 \) because there cannot be a negative number of candles or ornaments and the candles and ornaments together cannot be more than 8 items." - True.
Given that we need to select which of the statements is true:
The best conclusion is the last option, which encompasses the entire constraints clearly:
"The inequality that represents the constraint on number of candles is \( 0 \leq x \leq 8 \) and on the number of ornaments is \( 0 \leq y \leq 8 \) because there cannot be a negative number of candles or ornaments and the candles and ornaments together cannot be more than 8 items."