Question

1. one of the following two points lies in the solution set of the system of inequalities below. determine which point it is and explain why your choice lies in the solution set and the other does not.\\(x + y < 10\\)\\(y \geq \frac{2}{3}x - 2\\)\\((6,1)\\)\\((3,5)\\)

Explanation:

Step1: Check point (6,1) in \(x + y < 10\)

Substitute \(x = 6\), \(y = 1\): \(6 + 1 = 7 < 10\), satisfies first inequality.

Step2: Check point (6,1) in \(y \geq \frac{2}{3}x - 2\)

Calculate \(\frac{2}{3}(6)-2 = 4 - 2 = 2\). Check \(1 \geq 2\)? No, fails second inequality.

Step3: Check point (3,5) in \(x + y < 10\)

Substitute \(x = 3\), \(y = 5\): \(3 + 5 = 8 < 10\), satisfies first inequality.

Step4: Check point (3,5) in \(y \geq \frac{2}{3}x - 2\)

Calculate \(\frac{2}{3}(3)-2 = 2 - 2 = 0\). Check \(5 \geq 0\)? Yes, satisfies second inequality.

Answer:

The point \((3, 5)\) lies in the solution set because it satisfies both \(x + y < 10\) (\(3 + 5 = 8 < 10\)) and \(y \geq \frac{2}{3}x - 2\) (\(5 \geq \frac{2}{3}(3)-2 = 0\)). The point \((6, 1)\) fails \(y \geq \frac{2}{3}x - 2\) (since \(1
geq 2\)) so it is not in the solution set.