Question

kai and yaw are selling school spirit wear for a fundraiser. kai wants to sell six shirts and 4 pairs of socks and make at least $100. yaw wants to sell 5 shirts and 8 pairs of socks and hopes to earn at least $120. inequalities: what do the variables represent: list 2 possible solutions and their meaning:

Explanation:

Step1: Define variables

Let \(x\) be the price of a shirt and \(y\) be the price of a pair of socks.

Step2: Write inequalities

For Kai: \(6x + 4y\geq100\). Divide by \(2\) to simplify: \(3x + 2y\geq50\).
For Yaw: \(5x+8y\geq120\).

Step3: Find possible solutions

Let's assume \(x = 10\).
Substitute \(x = 10\) into \(3x + 2y\geq50\): \(3\times10+2y\geq50\), \(30 + 2y\geq50\), \(2y\geq20\), \(y\geq10\).
Check in \(5x + 8y\): \(5\times10+8y=50 + 8y\). If \(y = 10\), \(50+8\times10=130\geq120\).
Another solution: Let \(x = 12\).
Substitute into \(3x + 2y\geq50\): \(3\times12+2y\geq50\), \(36+2y\geq50\), \(2y\geq14\), \(y\geq7\).
Check in \(5x + 8y\): \(5\times12+8y=60 + 8y\). If \(y = 7\), \(60+8\times7=60 + 56=116\lt120\). If \(y = 8\), \(60+8\times8=60 + 64 = 124\geq120\).

Answer:

• Inequalities: \(6x + 4y\geq100\) (or \(3x + 2y\geq50\)) and \(5x + 8y\geq120\)
• Variables: \(x\) represents the price of a shirt, \(y\) represents the price of a pair of socks
• Possible solutions:
• Solution 1: \(x = 10\), \(y = 10\). Meaning: Each shirt is sold for \(\$10\) and each pair of socks is sold for \(\$10\). Kai would make \(6\times10+4\times10=\$100\) and Yaw would make \(5\times10 + 8\times10=\$130\)
• Solution 2: \(x = 12\), \(y = 8\). Meaning: Each shirt is sold for \(\$12\) and each pair of socks is sold for \(\$8\). Kai would make \(6\times12+4\times8=72 + 32=\$104\) and Yaw would make \(5\times12+8\times8=60+64=\$124\)