1. The problem is to determine if each point satisfies the inequality $x + 3y > 3$. Check each point: - For $(-3,4)$: $-3 + 3(4) = -3 + 12 = 9 > 3$ (True) - For $(5,5)$: $5 + 3(5) = 5 + 15 = 20 > 3$ (True) - For $(-5,8)$: $-5 + 3(8) = -5 + 24 = 19 > 3$ (True) - For $(3,-5)$: $3 + 3(-5) = 3 - 15 = -12 > 3$ (False) 2. Determine if each point satisfies $x > -3y + 4$. Check each point: - For $(6,-1)$: $6 > -3(-1) + 4$ implies $6 > 3 + 4 = 7$ (False) - For $(1,5)$: $1 > -3(5) + 4$ implies $1 > -15 + 4 = -11$ (True) - For $(0,5)$: $0 > -3(5) + 4$ implies $0 > -15 + 4 = -11$ (True) - For $(2,3)$: $2 > -3(3) + 4$ implies $2 > -9 + 4 = -5$ (True) 3. Determine if each point satisfies $2x - 3y \\leq 1$. Check each point: - For $(4,1)$: $2(4) - 3(1) = 8 - 3 = 5 \\leq 1$ (False) - For $(1,1)$: $2(1) - 3(1) = 2 - 3 = -1 \\leq 1$ (True) - For $(0,0)$: $2(0) - 3(0) = 0 \\leq 1$ (True) - For $(5,-3)$: $2(5) - 3(-3) = 10 + 9 = 19 \\leq 1$ (False) 4. Determine if each point satisfies $10 < -5x + 2y$. Check each point: - For $(-3,1)$: $10 < -5(-3) + 2(1) = 15 + 2 = 17$ (True) - For $(-5,4)$: $10 < -5(-5) + 2(4) = 25 + 8 = 33$ (True) - For $(-2,-2)$: $10 < -5(-2) + 2(-2) = 10 - 4 = 6$ (False) - For $(4,4)$: $10 < -5(4) + 2(4) = -20 + 8 = -12$ (False) 5. Determine if each point satisfies $4x + 3y \\geq 12$. Check each point: - For $(-4,-3)$: $4(-4) + 3(-3) = -16 - 9 = -25 \\geq 12$ (False) - For $(5,4)$: $4(5) + 3(4) = 20 + 12 = 32 \\geq 12$ (True) - For $(5,-1)$: $4(5) + 3(-1) = 20 - 3 = 17 \\geq 12$ (True) - For $(1,-6)$: $4(1) + 3(-6) = 4 - 18 = -14 \\geq 12$ (False)