The determinant is 0 because the first column is repeated.

|z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.

|z| = √(4^2 + (-3)^2) = √(16 + 9) = √25 = 5.

The modulus |z| = 4, as it is the coefficient in the polar form.

|z|^2 = 4^2 + (-3)^2 = 16 + 9 = 25.

The vertex can be found using x = -b/(2a) = 4. Substituting x = 4 into the equation gives y = -3.

The discriminant for x^2 + 4x + 5 is negative (16 - 20 < 0), indicating no real roots.

A purely imaginary number has no real part, so 0 + 5i is purely imaginary.

The roots are z = ±i.

log_a(b^c) = c * log_a(b) by the power rule of logarithms.

x + 2 > 1 => x > -1.

x + 2 > 7 => x > 5.

x + 5 > 2 => x > -3.

3x + 2 > 11 => 3x > 9 => x > 3.

3x + 2 ≥ 11 => 3x ≥ 9 => x ≥ 3.

x + 2 > 7 => x > 5.

4 - x < 1 => -x < -3 => x > 3

4 - x < 2 => -x < -2 => x > 2.

6x - 4 ≤ 2x + 8 => 4x ≤ 12 => x ≤ 3.

7x - 4 < 10 => 7x < 14 => x < 2.

x/3 + 2 < 5 => x/3 < 3 => x < 9.

x/3 - 2 < 1 => x/3 < 3 => x < 9.