Add 7 to both sides: 3x ≤ 9. Then divide by 3: x ≤ 3.

6 - 2x ≤ 0 => -2x ≤ -6 => x ≥ 3.

6 - 3x ≤ 0 => -3x ≤ -6 => x ≥ 2.

7 - 3x > 1 => -3x > -6 => x < 2.

8x + 1 ≤ 5 => 8x ≤ 4 => x ≤ 0.5.

x + 2 > 3 => x > 1.

The sum of the roots is given by -b/a = 12/3 = 4.

Using the binomial theorem, (1 + 2)^4 = C(4,0) * 1^4 * 2^0 + C(4,1) * 1^3 * 2^1 + C(4,2) * 1^2 * 2^2 + C(4,3) * 1^1 * 2^3 + C(4,4) * 1^0 * 2^4 = 1 + 8 + 24 + 32 + 16 = 81.

(1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i.

(1 + i)^4 = (√2 e^(iπ/4))^4 = 4 e^(iπ) = 4(-1) = -4.

Using the binomial theorem, (1 + 1)^10 = 2^10 = 1024.

Using the binomial theorem, (1 + 2)^10 = 3^10 = 59049.

cos(tan^(-1)(1)) = 1/√2

cos(tan^(-1)(3)) = 3/√10

cos^(-1)(-1/2) = 2π/3, since cos(2π/3) = -1/2.

i^4 = (i^2)^2 = (-1)^2 = 1.

For no real roots, the discriminant must be less than 0: k^2 - 4*1*16 < 0, which gives k < 8.

For both roots to be negative, k must be positive and k^2 > 36, thus k > 6.

The discriminant must be positive: k^2 - 4*1*9 > 0, which gives k < 6 or k > -6.

For no real roots, the discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => |k| < 8.

For no real roots, the discriminant must be less than zero: k^2 - 36 < 0, hence k < -6.

The coefficient of x^4 is C(6,4) * k^2 = 15k^2. Setting 15k^2 = 240 gives k^2 = 16, so k = 4 or -4.

Let θ = cos^(-1)(1/2). Then cos(θ) = 1/2, which corresponds to θ = π/3. Therefore, sin(θ) = sin(π/3) = √3/2.

sin(tan^(-1)(1)) = 1/√2, using the triangle with opposite and adjacent sides equal.

Using the right triangle definition, sin(tan^(-1)(x)) = opposite/hypotenuse = x/√(1+x^2).

Since π/3 is in the range of sin^(-1), sin^(-1)(sin(π/3)) = π/3.

Since sin(π/4) = 1/√2, sin^(-1)(1/√2) = π/4.

Since sin^2(θ) = 1 - cos^2(θ), we have sin^(-1)(√(1 - cos^2(θ))) = sin^(-1)(sin(θ)) = θ.

sin^(-1)(√3/2) = π/3 and sin^(-1)(1/2) = π/6. Therefore, π/3 + π/6 = π/2.

sin^(-1)(√3/2) = π/3.