$$ - 6 < x < 7 $$

$$ x < {1 \over 4} \text{ or } x > 3 $$

\begin{align} 9x^2 & \le 16 \\ 9x^2 - 16 & \le 0 \\ (3x)^2 - 4^2 & \le 0 \\ (3x + 4)(3x - 4) & \le 0 \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \end{align}

$$ -{4 \over 3} \le x \le {4 \over 3} $$

\begin{align} 14x^2 + 39x + 10 & \ge 0 \\ (2x + 5)(7x + 2) & \ge 0 \end{align}

$$ x \le -{5 \over 2} \text{ or } x \ge -{2 \over 7} $$

\begin{align} 8(x^2 + 3x - 1) & < (3x + 1)(x - 2) \\ 8x^2 + 24x - 8 & < 3x^2 - 6x + x - 2 \\ 8x^2 + 24x - 8 & < 3x^2 - 5x - 2 \\ 5x^2 + 29x - 6 & < 0 \\ (x + 6)(5x - 1) & < 0 \end{align}

$$ -6 < x < {1 \over 5} $$

\begin{align} (x + 4)^2 - 11(x + 4) + 18 & > 0 \\ (x + 4)(x + 4) - 11x - 44 + 18 & > 0 \\ x^2 + 4x + 4x + 16 - 11x - 44 + 18 & > 0 \\ x^2 - 3x - 10 & > 0 \\ (x + 2)(x - 5) & > 0 \end{align}

$$ x < -2 \text{ or } x > 5 $$

\begin{align} {81 - x^2 \over x^2} & \le 0 \\ \\ \text{Since } x^2 \ge 0 & \text{ for all real values of } x, \\ 81 - x^2 & \le 0 \\ x^2 - 81 & \ge 0 \\ x^2 - 9^2 & \ge 0 \\ (x + 9)(x - 9) & \ge 0 \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \end{align}

$$ x \le - 9 \text{ or } x \ge 9 $$

\begin{align} {81 - x^4 \over x^4} & \ge 0 \\ \\ \text{Since } x^4 \ge 0 & \text{ for all real values of } x, \\ 81 - x^4 & \ge 0 \\ x^4 - 81 & \le 0 \\ (x^2)^2 - 9^2 & \le 0 \\ (x^2 + 9)(x^2 - 9) & \le 0 \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \\ \\ \text{Since } x^2 + 9 \ge 9 & \text{ for all real values of } x, \\ x^2 - 9 & \le 0 \\ x^2 - 3^2 & \le 0 \\ (x + 3)(x - 3) & \le 0 \end{align}

$$ -3 \le x \le 3, x \ne 0 \phantom{0} \phantom{000000} \left[ {81 - x^4 \over x^4} \text{ is undefined if } x = 0 \right] $$

\begin{align} 8x^2 - 5x - 2 & < 2x(x - 2) \\ 8x^2 - 5x - 2 & < 2x^2 - 4x \\ 6x^2 - x - 2 & < 0 \\ (2x + 1)(3x - 2) & < 0 \end{align}

$$ -{1 \over 2} < x < {2 \over 3} $$

\begin{align} 2x^2 + 3x - 20 & < 0 \\ (x + 4)(2x - 5) & < 0 \end{align}

$$ -4 < x < {5 \over 2} $$

\begin{align} \implies (6x + 1)(3x - 4) & > 0 \\ 18x^2 - 24x + 3x - 4 & > 0 \\ 18x^2 - 21x - 4 & > 0 \\ -1(18x^2 - 21x - 4) & < -1(0) \\ -18x^2 + 21x + 4 & < 0 \\ 4 + 21x - 18x^2 & < 0 \end{align}

\begin{align} {1 \over 5}x + 1 & \ge -2 \\ {1 \over 5}x & \ge -3 \\ x & \ge 5(-3) \\ x & \ge -15 \phantom{0000000} [\text{First condition}] \\ \\ \\ 3 - 11x & < 4x^2 \\ 0 & < 4x^2 + 11x - 3 \\ 0 & < (x + 3)(4x - 1) \end{align}

$$ x < -3 \text{ or } x > {1 \over 4} \phantom{0000000} [\text{Second condition}] $$

$$ -15 \le x < -3 \text{ or } x > {1 \over 4} $$

\begin{align} x(x + 6) & < 2x^2 + 11x + 6 \\ x^2 + 6x & < 2x^2 + 11x + 6 \\ 0 & < x^2 + 5x + 6 \\ 0 & < (x + 3)(x + 2) \end{align}

\begin{align} x < -3 & \text{ or } x > -2 \phantom{000000} [\text{First condition}] \\ \\ \\ 2x^2 + 11x + 6 & < -8 \\ 2x^2 + 11x + 14 & < 0 \\ (2x + 7)(x + 2) & < 0 \end{align}

$$ - {7 \over 2} < x < -2 \phantom{000000} [\text{Second condition}] $$

$$ -{7 \over 2} < x < -3 $$

\begin{align} x^2 - (h + 1)x & + h + 9 = 0 \\ \\ b^2 - 4ac & = [-(h + 1)]^2 - 4(1)(h + 9) \\ & = (h + 1)(h + 1) - 4(h + 9) \\ & = h^2 + h + h + 1 - 4h - 36 \\ & = h^2 - 2h - 35 \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No real roots}] \\ h^2 - 2h - 35 & < 0 \\ (h + 5)(h - 7) & < 0 \end{align}

$$ -5 < h < 7 $$

\begin{align} {k + 8 \over x^2} + {6 \over x} + k & = 0 \\ {k + 8 \over x^2} + {6x \over x^2} + {kx^2 \over x^2} & = 0 \\ {k + 8 + 6x + kx^2 \over x^2} & = 0 \\ \\ k + 8 + 6x + kx^2 & = 0 \\ kx^2 + 6x + k + 8 & = 0 \\ \\ k & \ne 0 \phantom{000000} [\text{Or else equation only has 1 root}] \\ \\ b^2 - 4ac & = (6)^2 - 4(k)(k + 8) \\ & = 36 - 4k(k + 8) \\ & = 36 - 4k^2 - 32k \\ \\ b^2 - 4ac & > 0 \phantom{000000} [\text{Two distinct real roots}] \\ 36 - 4k^2 - 32k & > 0 \\ -4k^2 - 32k + 36 & > 0 \\ 4k^2 + 32k - 36 & < 0 \\ k^2 + 8k - 9 & < 0 \\ (k + 9)(k - 1) & < 0 \end{align}

\begin{align} - 9 & < k < 1, k \ne 0 \\ \\ \text{Least integer} & \text{ value of } k = -1 \end{align}

\begin{align} -kx^2 & + 5kx + (k - 4) \\ \\ -k & < 0 \phantom{000000} [y = -kx^2 + 5kx + (k - 4) \text{ is maximum curve } \cap] \\ k & > 0 \phantom{000000} [\text{Condition 1}] \\ \\ \\ b^2 - 4ac & = (5k)^2 - 4(-k)(k - 4) \\ & = 25k^2 + 4k(k - 4) \\ & = 25k^2 + 4k^2 - 16k \\ & = 29k^2 - 16k \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No real roots for } -kx^2 + 5kx + (k - 4) = 0] \\ 29k^2 - 16k & < 0 \\ k(29k - 16) & < 0 \end{align}

\begin{align} 0 & < k < {16 \over 29} (\approx 0.55172) \\ \\ \\ \text{Irrational} & \text{ value of } k = {\pi \over 10} \end{align}

\begin{align} y & = x^2 + ax + 16 \phantom{00} \text{--- (1)} \\ \\ y + 2x + a & = 0 \\ y & = -2x - a \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ -2x - a & = x^2 + ax + 16 \\ 0 & = x^2 + ax + 2x + 16 + a \\ 0 & = x^2 + (a + 2)x + 16 + a \\ \\ \\ b^2 - 4ac & = (a + 2)^2 - 4(1)(16 + a) \\ & = (a + 2)(a + 2) - 4(16 + a) \\ & = a^2 + 2a + 2a + 4 - 64 - 4a \\ & = a^2 - 64 \\ \\ b^2 - 4ac & \le 0 \phantom{000000} [\text{Line meets curve once or not at all}] \\ a^2 - 64 & \le 0 \\ a^2 - (\sqrt{64})^2 & \le 0 \\ (a + \sqrt{64})(a - \sqrt{64}) & \le 0 \end{align}

\begin{align} -\sqrt{64} & \le a \le \sqrt{64} \\ - \sqrt{4}\sqrt{15} & \le a \le \sqrt{4} \sqrt{15} \\ -2 \sqrt{15} & \le a \le 2 \sqrt{15} \phantom{00} (\text{Shown}) \end{align}

\begin{align} y & = {x^2 + 2x + 13 \over x - 1} \\ y(x - 1) & = x^2 + 2x + 13 \\ yx - y & = x^2 + 2x + 13 \\ 0 & = x^2 + 2x - yx + y + 13 \\ 0 & = x^2 + (2 - y)x + (y + 13) \\ \\ b^2 - 4ac & = (2 - y)^2 - 4(1)(y + 13) \\ & = (2 - y)(2 - y) - 4(y + 13) \\ & = 4 - 2y - 2y + y^2 - 4y - 52 \\ & = y^2 - 8y - 48 \\ \\ b^2 - 4ac & \ge 0 \\ y^2 - 8y - 48 & \ge 0 \\ (y + 4)(y - 12) & \ge 0 \end{align}

$$ y \le -4 \text{ or } y \ge 12 $$