Question

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) 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. 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. 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. 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.

Explanation:

Step1: Analyze the number - of - items constraint

Since Sylvia buys a total of 8 items, \(x + y=8\), which can be rewritten as \(y = 8 - x\). The number of candles \(x\) and the number of ornaments \(y\) cannot be negative. Also, from \(y=8 - x\geq0\), we get \(x\leq8\), and from \(x = 8 - y\geq0\), we get \(y\leq8\).

Step2: Consider the non - negativity constraints

The number of objects (candles and ornaments) cannot be negative. So \(x\geq0\) and \(y\geq0\). Combining with \(x + y = 8\), we have \(0\leq x\leq8\) and \(0\leq y\leq8\).

Answer:

The inequality that represents the constraint on number of candles is \(0\leq x\leq8\) and on the number of ornaments is \(0\leq y\leq8\) because there cannot be a negative number of candles or ornaments and the candles and ornaments together cannot be more than 8 items.