(a)

$$ -2 < x < 1 $$

(b)

$$ -3 \le x \le 4 $$

(c)

$$ x < -{3 \over 2} \text{ or } x > 2 $$

(d)

$$ x \le 0 \text{ or } x \ge 5 $$

(e)

\begin{align} x^2 - 4x & \le 0 \\ x(x - 4) & \le 0 \end{align} $$ 0 \le x \le 4 $$

(f)

\begin{align*} x^2 - 4 & < 0 \\ x^2 - 2^2 & < 0 \\ (x + 2)(x - 2) & < 0 \end{align*} $$ -2 < x < 2 $$

\begin{align} 2x^2 - 4x - 3 & > x \\ 2x^2 - 5x - 3 & > 0 \\ (2x + 1)(x - 3) & > 0 \end{align}
$$ x < -{1 \over 2} \text{ or } x > 3 $$

(a)

\begin{align} x(x - 2) & < 3 \\ x^2 - 2x & < 3 \\ x^2 - 2x - 3 & < 0 \\ (x + 1)(x - 3) & < 0 \end{align}
$$ -1 < x < 3 $$

(b)

\begin{align} x^2 & > 4x + 12 \\ x^2 - 4x - 12 & > 0 \\ (x + 2)(x - 6) & > 0 \end{align}
$$ x < - 2 \text{ or } x > 6 $$

(c)

\begin{align} 4x(x + 1) & \le 3 \\ 4x^2 + 4x & \le 3 \\ 4x^2 + 4x - 3 & \le 0 \\ (2x + 3)(2x - 1) & \le 0 \end{align}
$$ -{3 \over 2} \le x \le {1 \over 2} $$

(d)

\begin{align} (1 - x)^2 & \ge 17 - 2x \\ (1)^2 - 2(1)(x) + (x)^2 & \ge 17 - 2x \\ 1 - 2x + x^2 & \ge 17 - 2x \\ 1 - 17 - 2x + 2x + x^2 & \ge 0 \\ - 16 + x^2 & \ge 0 \\ x^2 - 16 & \ge 0 \\ (x + 4)(x - 4) & \ge 0 \end{align}
$$ x \le - 4 \text{ or } x \ge 4 $$

(e)

\begin{align} (x + 2)^2 & < x(4 - x) + 40 \\ x^2 + 2(x)(2) + 2^2 & < 4x - x^2 + 40 \\ x^2 + 4x + 4 & < 4x - x^2 + 40 \\ x^2 + x^2 + 4x - 4x + 4 - 40 & < 0 \\ 2x^2 - 36 & < 0 \\ x^2 - 18 & < 0 \\ x^2 - (\sqrt{18})^2 & < 0 \\ (x + \sqrt{18})(x - \sqrt{18}) & < 0 \end{align}
$$ - \sqrt{18} < x < \sqrt{18} $$

\begin{align} S & > 3800 \\ \\ \text{Since } S = 600 & + 520T - 20T^2, \\ \\ 600 + 520T - 20T^2 & > 3800 \\ 600 - 3800 + 520T - 20T^2 & > 0 \\ -3200 + 520T - 20T^2 & > 0 \\ -20T^2 + 520T - 3200 & > 0 \\ 20T^2 - 520T + 3200 & < 0 \\ T^2 - 26T + 160 & < 0 \\ (T - 10)(T - 16) & < 0 \end{align}
\begin{align} 10 < \phantom{.} & T < 16 \\ \\ \therefore \text{Between } 10^\circ & \text{C and } 16^\circ \text{C} \end{align}

\begin{align} C & < 150 \\ \\ \text{Since } & C = 5x^2 - 38x + 222, \\ 5x^2 - 38x + 222 & < 150 \\ 5x^2 - 38x + 222 - 150 & < 0 \\ 5x^2 - 38x + 72 & < 0 \\ (5x - 18)(x - 4) & < 0 \end{align}
$$ 3{3 \over 5} < x < 4 $$

(a)

\begin{align} x^2 - px + p & = 0 \\ \\ [a = 1, b & = -p, c = p] \\ \\ b^2 - 4ac & > 0 \\ (-p)^2 - 4(1)(p) & > 0 \\ p^2 - 4p & > 0 \\ p(p - 4) & > 0 \end{align}
$$ p < 0 \text{ or } p > 4 $$

(b)

\begin{align} 9x^2 + 2px + 1 & = 0 \\ \\ [a = 9, b & = 2p, c = 1] \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{2 real distinct roots or 2 real equal roots}]\\ (2p)^2 - 4(9)(1) & \ge 0 \\ 4p^2 - 36 & \ge 0 \\ p^2 - 9 & \ge 0 \\ (p + 3)(p - 3) & \ge 0 \end{align}
$$ p \le - 3 \text{ or } p \ge 3 $$

(c)

\begin{align} px^2 - 2x + 2p + 1 & = 0 \\ \\ [a = p, b & = -2, c = 2p + 1] \\ \\ b^2 - 4ac & < 0 \\ (-2)^2 - 4(p)(2p + 1) & < 0 \\ 4 - 4p (2p + 1) & < 0 \\ 4 - 8p^2 - 4p & < 0 \\ -8p^2 - 4p + 4 & < 0 \\ 8p^2 + 4p - 4 & > 0 \\ 2p^2 + p - 1 & > 0 \\ (p + 1)(2p - 1) & > 0 \end{align}
$$ p < -1 \text{ or } p > {1 \over 2} $$

(d)

\begin{align} 2x^2 - 2px + p^2 + p & = 0 \\ \\ [a = 2, b & = -2p, c = p^2 + p] \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{2 real distinct roots or 2 real equal roots}] \\ (-2p)^2 - 4(2)(p^2 + p) & \ge 0 \\ 4p^2 - 8(p^2 + p) & \ge 0 \\ 4p^2 - 8p^2 - 8p & \ge 0 \\ -4p^2 - 8p & \ge 0 \\ \\ 4p^2 + 8p & \le 0 \\ p^2 + 2p & \le 0 \\ p(p + 2) & \le 0 \end{align}
$$ -2 \le p \le 0 $$

(a)

\begin{align} -2 < \phantom{.} & x < 4 \\ \\ \text{Working } & \text{backwards}, \\ \\ (x + 2)(x - 4) & < 0 \\ x^2 - 4x + 2x - 8 & < 0 \\ x^2 - 2x - 8 & < 0 \\ x^2 - 2x & < 8 \\ \\ \therefore a = -2, & \phantom{.} b = 8 \end{align}

(b)

\begin{align} x < -2 \text{ or } & x > 3 \\ \\ \text{Working } & \text{backwards}, \\ \\ (x + 2)(x - 3) & > 0 \\ x^2 - 3x + 2x - 6 & > 0 \\ x^2 - x - 6 & > 0 \\ x^2 - 6 & > x \\ 2x^2 - 12 & > 2x \\ \\ \therefore a = -12, & \phantom{.} b = 2 \end{align}

\begin{align} 3kx^2 + (k - 5)x & = 5x^2 + 2 \\ 3kx^2 - 5x^2 + (k - 5)x - 2 & = 0 \\ (3k - 5)x^2 + (k - 5)x - 2 & = 0 \\ \\ [a = 3k - 5, b & = k - 5, c = - 2] \\ \\ b^2 - 4ac & < 0 \\ (k - 5)^2 - 4(3k - 5)(-2) & < 0 \\ k^2 - 2(k)(5) + 5^2 + 8(3k - 5) & < 0 \\ k^2 - 10k + 25 + 24k - 40 & < 0 \\ k^2 + 14k - 15 & < 0 \\ (k + 15)(k - 1) & < 0 \end{align}
$$ -15 < k < 1 $$

The equation can have 1 real root or 2 real roots, thus b² - 4ac ≥ 0

\begin{align} px^2 - p + 10 & = 2(p + 2)x \\ px^2 - p + 10 & = (2p + 4)x \\ px^2 - (2p + 4)x + 10 - p & = 0 \\ px^2 + (-2p - 4)x + 10 - p & = 0 \\ \\ [a = p, b & = -2p - 4, c = 10 - p] \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{2 real distinct roots or 2 real equal roots}] \\ (-2p - 4)^2 - 4(p)(10 - p) & \ge 0 \\ (-2p)^2 - 2(-2p)(4) + (4)^2 - 4p(10 - p) & \ge 0 \\ 4p^2 + 16p + 16 - 40p + 4p^2 & \ge 0 \\ 8p^2 - 24p + 16 & \ge 0 \\ p^2 - 3p + 2 & \ge 0 \\ (p - 1)(p - 2) & \ge 0 \end{align}
$$ x \le 1 \text{ or } x \ge 2 $$ $$ \therefore p \text{ cannot lie between 1 and 2} $$

\begin{align} (p + 2)x^2 - 12x + 2(p - 1) & = 0 \\ (p + 2)x^2 - 12x + 2p - 2 & = 0 \\ \\ [a = p + 2, b & = -12, c = 2p - 2] \\ \\ b^2 - 4ac & > 0 \\ (-12)^2 - 4(p + 2)(2p - 2) & > 0 \\ 144 - 4(2p^2 - 2p + 4p - 4) & > 0 \\ 144 - 4(2p^2 + 2p - 4) & > 0 \\ 144 - 8p^2 - 8p + 16 & > 0 \\ -8p^2 - 8p + 160 & > 0 \\ \\ 8p^2 + 8p - 160 & < 0 \\ p^2 + p - 20 & < 0 \\ (p + 5)(p - 4) & < 0 \end{align}
$$ -5 < p < 4 $$

\begin{align} y & = 2x + c \phantom{000} \text{ --- (1)} \\ \\ 2xy + 6 & = 0 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 2x(2x + c) + 6 & = 0 \\ 4x^2 + 2cx + 6 & = 0 \\ \\ [a = 4, b & = 2c, c = 6] \\ \\ b^2 - 4ac & < 0 \\ (2c)^2 - 4(4)(6) & < 0 \\ 4c^2 - 96 & < 0 \\ c^2 - 24 & < 0 \\ c^2 - (\sqrt{24})^2 & < 0 \\ (c + \sqrt{24})(c - \sqrt{24}) & < 0 \end{align}
$$ -\sqrt{24} < c < \sqrt{24} $$

\begin{align} 2x - y & = k \\ -y & = k - 2x \\ y & = 2x - k \phantom{000} \text{ --- (1)} \\ \\ x^2 - xy + y^2 & = 1 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x^2 - x(2x - k) + (2x - k)^2 & = 1 \\ x^2 - 2x^2 + kx + (2x)^2 - 2(2x)(k) + (k)^2 & = 1 \\ - x^2 + kx + 4x^2 - 4kx + k^2 & = 1 \\ 3x^2 - 3kx + k^2 - 1 & = 0 \\ \\ [a = 3, b & = -3k, c = k^2 - 1] \\ \\ b^2 - 4ac & > 0 \\ (-3k)^2 - 4(3)(k^2 - 1) & > 0 \\ 9k^2 - 12(k^2 - 1) & > 0 \\ 9k^2 - 12k^2 + 12 & > 0 \\ -3k^2 + 12 & > 0 \\ \\ 3k^2 - 12 & < 0 \\ k^2 - 4 & < 0 \\ (k + 2)(k - 2) & < 0 \end{align}
$$ -2 < k < 2 $$

(i)

\begin{align} d & \le 80 \\ \\ \text{Since } & d = 0.15v^2 + v, \\ 0.15v^2 + v & \le 80 \\ 0.15v^2 + v - 80 & \le 0 \\ 1.5v^2 + 10v - 800 & \le 0 \phantom{0000000} [\text{Multiply by 10}] \\ \\ 3v^2 + 20v - 1600 & \le 0 \text{ (Shown)} \end{align}

(ii)

\begin{align} 3v^2 + 20v - 1600 & \le 0 \\ (3v + 80)(v - 20) & \le 0 \end{align}
$$ - {80 \over 3} \le v \le 20 $$ $$ \therefore \text{Maximum speed} = 20 \text{ m/s} $$

\begin{align} \text{Let } A \text{ denote } & \text{the surface area of the cylinder}. \\ \\ A & = 2\pi r^2 + 2\pi r h \\ \\ \text{When } & h = 6, \\ A & = 2\pi r^2 + 2\pi r (6) \\ A & = 2\pi r^2 + 12\pi r \\ \\ \\ \text{Since } & A > 224\pi, \\ 2\pi r^2 + 12\pi r & > 224\pi \\ 2r^2 + 12r & > 224 \\ 2r^2 + 12r - 224 & > 0 \\ r^2 + 6r - 112 & > 0 \\ (r + 14)(r - 8) & > 0 \end{align}
$$ r < - 14 \text{ (Reject)} \text{ or } r > 8 $$

(i)(a)

\begin{align} 5x - 7 & \ge 3(x - 5) \\ 5x - 7 & \ge 3x - 15 \\ 5x - 3x & \ge -15 + 7 \\ 2x & \ge -8 \\ x & \ge -4 \end{align}

(i)(b)

\begin{align} x(2x + 1) & > 6 \\ 2x^2 + x & > 6 \\ 2x^2 + x - 6 & > 0 \\ (x + 2)(2x - 3) & > 0 \end{align}
$$ x < -2 \text{ or } x > 1.5 $$

(ii) Consolidate the number lines from (i)(a) and (i)(b) as a single number line:

$$ -4 \le x < -2 \text{ or } x > 1.5 $$

\begin{align} 2 < \phantom{.} & x < k \\ \\ \text{Working } & \text{backwards,} \\ \\ (x - 2)(x - k) & < 0 \\ x^2 - kx - 2x + 2k & < 0 \\ 2x^2 - 2kx - 4x + 4k & < 0 \\ 2x^2 + (-2k - 4)x + 4k & < 0 \\ \\ \\ \therefore y = 2x^2 + & (-2k - 4)x + 4k \\ \\ \text{By comparison with } y & = 2x^2 + px + 16, \\ \\ 4k & = 16 \\ k & = {16 \over 4} \\ & = 4 \\ \\ \\ -2k - 4 & = p \\ -2(4) - 4 & = p \\ -12 & = p \end{align}

\begin{align} y = (k - 6)x^2 - & 8x + k \\ \\ [a= k - 6, b & = -8, c = k] \\ \\ b^2 - 4ac & > 0 \\ (-8)^2 - 4(k - 6)(k) & > 0 \\ 64 - 4k(k - 6) & > 0 \\ 64 - 4k^2 + 24k & > 0 \\ -4k^2 + 24k + 64 & > 0 \\ \\ 4k^2 - 24k - 64 & < 0 \\ k^2 - 6k - 16 & < 0 \\ (k + 2)(k - 8) & < 0 \end{align}
\begin{align} -2 < \phantom{.} & k < 8 \\ \\ \text{Since curve has a minimum point,} & \phantom{0} \text{coefficient of } x^2 \text{ must be positive.} \\ \\ k - 6 & > 0 \\ k & > 6 \\ \\ \therefore 6 < \phantom{.} & k < 8 \end{align}

(a)

\begin{align} x^2 + 2x & < 0 \\ x(x + 2) & < 0 \end{align} $$ -2 < x < 0 $$

\begin{align} x^2 - x & > 2 \\ x^2 - x - 2 & > 0 \\ (x + 1)(x - 2) & > 0 \end{align} $$ x < -1 \text{ or } x > 2 $$

Illustrate both inequalities on a number line:

$$ \therefore -2 < x < -1 $$

(b)

\begin{align} x^2 & \ge 4 \\ x^2 - 4 & \ge 0 \\ (x + 2)(x - 2) & \ge 0 \end{align} $$ x \le - 2 \text{ or } x \ge 2 $$

\begin{align} x^2 - 6 & > 5x \\ x^2 - 5x - 6 & > 0 \\ (x + 1)(x - 6) & > 0 \end{align} $$ x < -1 \text{ or } x > 6 $$

Illustrate both inequalities on a number line:

$$ \therefore x \le -2 \text{ or } x > 6 $$

(c)

\begin{align} 0 < x(x - & 3) \le x \\ \\ 0 < x(x - 3) \phantom{000} & \text{or} \phantom{000} x(x - 3) \le x \end{align}
Solve the first inequality:
$$ 0 < x(x - 3) $$ $$ x < 0 \text{ or } x > 3 $$

Solve the second inequality:
\begin{align} x(x - 3) & \le x \\ x^2 - 3x & \le x \\ x^2 - 4x & \le 0 \\ x(x - 4) & \le 0 \end{align} $$ 0 \le x \le 4 $$

Illustrate both inequalities on a number line:

$$\therefore 3 < x \le 4 $$

(d)

\begin{align} x \le \phantom{.} & x^2 < 9 \\ \\ x \le x^2 \phantom{000}&\text{or}\phantom{000} x^2 < 9 \end{align}
Solve the first inequality:
\begin{align} x & \le x^2 \\ 0 & \le x^2 - x \\ 0 & \le x(x - 1) \end{align} $$ x \le 0 \text{ or } x \ge 1 $$

Solve the second inequality:
\begin{align} x^2 & < 9 \\ x^2 - 9 & < 0 \\ (x + 3)(x - 3) & < 0 \end{align} $$ -3 < x < 3 $$

Illustrate both inequalities on a number line:

$$ \therefore -3 < x \le 0 \text{ or } 1 \le x < 3 $$

(i)

\begin{align} 1 + 2x + {3 \over 2}(x - 1)x & = 100000 \\ 2 + 4x + 3(x - 1)x & = 200000 \\ 2 + 4x + 3x(x - 1) & = 200000 \\ 2 + 4x + 3x^2 - 3x & = 200000 \\ 3x^2 + 4x - 3x + 2 - 200000 & = 0 \\ 3x^2 + x - 199998 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-1 \pm \sqrt{1^2 - 4(3)(-199998)} \over 2(3)} \\ & = {-1 \pm \sqrt{2399977} \over 6} \\ & = 258.0309 \text{ or } -258.3643 \\ & \approx 258.03 \text{ or } -258.36 \end{align}

(ii)

\begin{align} 1 + 2x + {3 \over 2}(x - 1)x & \le 100000 \\ & . \\ & . \phantom{000} [\text{Same steps as part (i)}] \\ & . \\ \\ 3x^2 + x - 199998 & \le 0 \end{align}
Use the solutions from part (i):
$$ -258.36 \le x \le 258.03 $$

(iii)

\begin{align} \text{Since there can only be a } & \text{maximum of 100000 lines}, \\ \\ 1 + 2n + {3 \over 2}n(n - 1) & \le 100000 \\ & . \\ & . \phantom{000} [\text{Same steps as part (ii)}] \\ & . \\ \\ 3n^2 + n - 199998 & \le 0 \end{align}
Use the solutions from part (i):
$$ -258.36 \le n \le 258.03 $$ $$ \therefore \text{Maximum no. of users} = 258.03 \approx 258 $$