Sylvia spends $50 at the craft show on candles and ornaments. The candles cost $5 each, and the ornaments cost $7 each. Sylvia buys a total of 8 items at the craft show.

Let x=
number of candles and y=
number of ornaments. Which of the following statements is true?

(1 point)
Responses

The inequality that represents the constraint on the number of candles is x≤8
and on the number of ornaments is y≤8
because the candles and ornaments together cannot be more than 8 items.
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.

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.
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.

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 cannot be a negative number of candles or ornaments.
The inequality that represents the constraint on the number of candles is x is greater than or equal to 0 and on the number of ornaments is y is greater than or equal to 0 because there cannot be a negative number of candles or ornaments.

The inequality that represents the constraint on number of candles is 0≤x≤8
and on the number of ornaments is 0≤y≤8
because there cannot be a negative number of candles or ornaments and the candles and ornaments together cannot be more than 8 items.

1 answer

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:

  1. Sylvia spent a total of $50 on candles and ornaments.
  2. The cost of candles is $5 each, and the cost of ornaments is $7 each.
  3. The total number of items (candles and ornaments) she bought is 8.

From this information, we can identify the following constraints:

  • Total cost constraint: \[ 5x + 7y \leq 50 \]

  • Total items constraint: \[ x + y = 8 \]

  • Non-negativity constraint: \[ x \geq 0 \quad \text{and} \quad y \geq 0 \]

  • 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:

  1. "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

  2. "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.

  3. "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.

  4. "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.

  5. "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.

  6. "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."