(a)

\begin{align} {7 - x \over 4} & \le {1 - 2x \over 6} \\ {3(7 - x) \over 12} & \le {2(1 - 2x) \over 12} \\ 3(7 - x) & \le 2(1 - 2x) \\ 21 - 3x & \le 2 - 4x \\ -3x + 4x & \le 2 - 21 \\ x & \le -19 \end{align}

(b)

\begin{align} {x + 3 \over 2} & < 1 - {2x - 1 \over 5} \\ {5(x + 3) \over 10} & < {10 \over 10} - {2(2x - 1) \over 10} \\ {5(x + 3) \over 10} & < {10 - 2(2x - 1) \over 10} \\ 5(x + 3) & < 10 - 2(2x - 1) \\ 5x + 15 & < 10 - 4x + 2 \\ 5x + 4x & < 10 + 2 - 15 \\ 9x & < -3 \\ x & < {-3 \over 9} \\ x & < -{1 \over 3} \end{align}

(a)

\begin{align} -12 & < 3x - 8 &&& 3x - 8 & < 13 \\ -3x & < -8 + 12 &&& 3x & < 13 + 8 \\ -3x & < 4 &&& 3x & < 21 \\ x & > {4 \over -3} &&& x & < {21 \over 3} \\ x & > -1{1 \over 3} &&& x & < 7 \end{align}
$$ -1{1 \over 3} < x < 7 $$

(b)

\begin{align} -1{1 \over 3} < x < 7 & \\ \\ \text{Smallest integer value of } x & = -1 \\ \\ \text{Greatest integer value of } x & = 6 \end{align}

\begin{align} 7 - 3(3 - 2x) & < 8x + 11 &&& 8x + 11 & < 4x + 3 \\ 7 - 9 + 6x & < 8x + 11 &&& 8x - 4x & < 3 - 11 \\ 6x - 8x & < 11 - 7 + 9 &&& 4x & < -8 \\ -2x & < 13 &&& x & < {-8 \over 4} \\ x & > {13 \over -2} &&& x & < -2 \\ x & > -6{1 \over 2} \end{align}
\begin{align} -6{1 \over 2} & \phantom{.} < x < -2 \\ \\ \text{Integers: } & -6, -5, -4, -3 \end{align}

(a)

\begin{align} -x & \le {3x - 2 \over 4} &&& {3x - 2 \over 4} & \le 13 \\ {-4x \over 4} & \le {3x - 2 \over 4} &&& {3x - 2 \over 4} & \le {52 \over 4} \\ -4x & \le 3x - 2 &&& 3x - 2 & \le 52 \\ -4x - 3x & \le -2 &&& 3x & \le 52 + 2 \\ -7x & \le -2 &&& 3x & \le 54 \\ x & \ge {-2 \over -7} &&& x & \le {54 \over 3} \\ x & \ge {2 \over 7} &&& x & \le 18 \end{align}
$$ {2 \over 7} \le x \le 18 $$

(b)(i) A rational number can be expressed as a fraction

$$ {2 \over 7} $$

(b)(ii) A prime number is a positive integer with only two factors, one and itself.

$$ 17 $$

(b)(iii) Square number (or perfect square): 1², 2², 3², 4², ...

$$ 4^2 = 16 $$

(a)

\begin{align} 3(x - 1) & < 2(9 + 5x) &&& {2x \over 5} - {3x \over 4} & < 1 - {x \over 2} \\ 3x - 3 & < 18 + 10x &&& {8x \over 20} - {15x \over 20} & < {20 \over 20} - {10x \over 20} \\ 3x - 10x & < 18 + 3 &&& {8x - 15x \over 20} & < {20 - 10x \over 20} \\ -7x & < 21 &&& 8x - 15x & < 20 - 10x \\ x & > {21 \over -7} &&& -7x & < 20 - 10x \\ x & > -3 &&& -7x + 10x & < 20 \\ & &&& 3x & < 20 \\ & &&& x & < {20 \over 3} \\ & &&& x & < 6{2 \over 3} \end{align}
$$ -3 < x < 6{2 \over 3} $$

(b)

\begin{align} -8 & \le 5x + 2 &&& 5x + 2 & < 37 \\ -5x & \le 2 + 8 &&& 5x & < 37 - 2 \\ -5x & \le 10 &&& 5x & < 35 \\ x & \ge {10 \over -5} &&& x & < {35 \over 5} \\ x & \ge -2 &&& x & < 7 \end{align}
$$ -2 \le x < 7 $$

(c)

\begin{align} 13 - (1 - x) & < 2x &&& 2x & \le 6(7 - x) \\ 13 - 1 + x & < 2x &&& 2x & \le 42 - 6x \\ 12 + x & < 2x &&& 2x + 6x & \le 42 \\ x - 2x & < -12 &&& 8x & \le 42 \\ -x & < -12 &&& x & \le {42 \over 8} \\ x & > 12 &&& x & \le 5.25 \end{align}

$$ \text{No solutions} $$

(a)

\begin{align} x + y & = \underbrace{2}_\text{Largest} + \underbrace{3}_\text{Largest} \\ & = 5 \end{align}

(b)

\begin{align} x - y & = \underbrace{2}_\text{Largest} - \underbrace{ (-7)}_\text{Smallest} \\ & = 9 \end{align}

(c)

\begin{align} xy & = (2)(-7) \\ & = -14 \end{align}

(d) The square of a non-zero number will be a positive number, i.e. (-3)² = 9

\begin{align} x^2 - y^2 & = \underbrace{ (0)^2 }_\text{Smallest} - \underbrace{ (-7)^2 }_\text{Largest} \\ & = -49 \end{align}

(a)

\begin{align} {9 + x \over 5} & < {5 + x \over 1} &&& 5 + x & \le 6 \\ {9 + x \over 5} & < {5(5 + x) \over 5} &&& x & \le 6 - 5 \\ 9 + x & < 5(5 + x) &&& x & \le 1 \\ 9 + x & < 25 + 5x \\ x - 5x & < 25 - 9 \\ -4x & < 16 \\ x & > {16 \over -4} \\ x & > -4 \end{align}
$$ -4 < x \le 1 $$

(b)

\begin{align} \text{Smallest integer value of } x^2 & = (0)^2 \\ & = 0 \\ \\ \text{Largest integer value of } x^2 & = (-3)^2 \\ & = 9 \end{align}

(a)

\begin{align} v & = u + at \\ 30 & = 18 + a(4) \\ 30 & = 18 + 4a \\ 30 - 18 & = 4a \\ 12 & = 4a \\ {12 \over 4} & = a \\ 3 & = a \end{align}

(b)

\begin{align} v & = u + at \\ v & = 10 + 3t \end{align} \begin{align} 10 + 3t & > 16 &&& 10 + 3t & < 49 \\ 3t & > 16 - 10 &&& 3t & < 49 - 10 \\ 3t & > 6 &&& 3t & < 39 \\ t & > {6 \over 3} &&& t & < {39 \over 3} \\ t & > 2 &&& t & < 13 \end{align}
$$ 2 < t < 13 $$

\begin{align} \text{Let } x \text{ represent } & \text{the number of rental days} \\ \\ \text{Cost (Company } A) & = 45 \times x \\ & = \$ 45x \\ \\ \text{Cost (Company } B) & = 75 + 38 \times x \\ & = \$ (75 + 38x) \\ \\ 75 + 38x & < 45x \\ 38x - 45x & < -75 \\ -7x & < - 75 \\ x & > {-75 \over -7} \\ x & > 10{5 \over 7} \\ \\ \text{Minimum no. of days} & = 11 \end{align}

\begin{align} \text{Let } x \text{ represent time taken for} & \text{ a man's haircut (in mins)} \\ \\ \text{Time taken for woman's haircut} & = (x + 20) \text{ mins} \\ \\ \text{Total time} & = 8x + 5(x + 20) \\ & = 8x + 5x + 100 \\ & = (13x + 100) \text{ mins} \end{align}
\begin{align} 13x + 100 & > 295 &&& 13x + 100 & \le 503 \\ 13x & > 295 - 100 &&& 13x & \le 503 - 100 \\ 13x & > 195 &&& 13x & \le 403 \\ x & > {195 \over 13} &&& x & \le {403 \over 13} \\ x & > 15 &&& x & \le {31} \end{align}
\begin{align} & 15 < x \le 31 \\ \\ 15 \text{ mins} & < \text{Time taken (man)} \le 31 \text{ mins} \end{align}

\begin{align} \text{Marks scored from correct answers} & = (n + 8) \times 3 \\ & = 3(n + 8) \\ & = 3n + 24 \\ \\ \text{No. of wrong answers} & = 2n - (n + 8) \\ & = 2n - n - 8 \\ & = n - 8 \\ \\ \text{Marks deducted from wrong answers} & = -1 \times (n - 8) \\ & = -n + 8 \\ \\ \text{Total marks} & = 3n + 24 - n + 8 \\ & = 2n + 32 \end{align}
\begin{align} 2n + 32 & > 40 &&& 2n + 32 & < 50 \\ 2n & > 40 - 32 &&& 2n & < 50 - 32 \\ 2n & > 8 &&& 2n & < 18 \\ n & > {8 \over 2} &&& n & < {18 \over 2} \\ n & > 4 &&& n & < 9 \end{align}
$$ 4 < n < 9 $$