(a)

\begin{align} 4x & < 2x + 3 \\ 4x - 2x & < 3 \\ 2x & < 3 \\ x & < {3 \over 2} \end{align}

(b)

\begin{align} 3x & > 7x + 10 \\ 3x - 7x & > 10 \\ -4x & > 10 \\ x & < {10 \over -4} \\ x & < -2.5 \end{align}

(c)

\begin{align} 5x & \ge 2(2x + 3) \\ 5x & \ge 4x + 6 \\ 5x - 4x & \ge 6 \\ x & \ge 6 \end{align}

(d)

\begin{align} 2x + 5 & \le 6x - 13 \\ 2x - 6x & \le -13 - 5 \\ - 4x & \le -18 \\ x & \ge {-18 \over -4} \\ x & \ge 4.5 \end{align}

\begin{align} 2x - 3 & \le 7 &&& 2x + 1 & \ge - 3x - 4 \\ 2x & \le 7 + 3 &&& 2x + 3x & \ge - 4 - 1 \\ 2x & \le 10 &&& 5x & \ge - 5 \\ x & \le {10 \over 2} &&& x & \ge {-5 \over 5} \\ x & \le 5 &&& x & \ge -1 \end{align}
$$ -1 \le x \le 5 $$

\begin{align} {y - 2 \over 3} & < {2y + 1 \over 5} &&& {2y + 1 \over 5} & \le 3 \\ {5(y - 2) \over 15} & < {3(2y + 1) \over 15} &&& {2y + 1 \over 5} & \le {15 \over 5} \\ 5(y - 2) & < 3(2y + 1) &&& 2y + 1 & \le 15 \\ 5y - 10 & < 6y + 3 &&& 2y & \le 15 - 1 \\ 5y - 6y & < 3 + 10 &&& 2y & \le 14 \\ -y & < 13 &&& y & \le {14 \over 2} \\ y & > -13 &&& y & \le 7 \end{align}
$$ -13 < y \le 7 $$

\begin{align} 20 \text{ cents} & = \$ 0.20 \\ \\ 50 \text{ cents} & = \$ 0.50 \\ \\ \text{Let } x \text{ represent } & \text{the number of 50-cent coins} \\ \\ \text{No. of 20-cent coins} & = 48 - x \end{align}
\begin{align} \text{No. of 20 cent coins} & \ge 2 \times \text{No. of 50 cent coins} &&& \text{Total value of coins} & > \$12 \\ 48 - x & \ge 2 \times x &&& 0.20(48 - x) + 0.50(x) & > 12 \\ 48 - x & \ge 2x &&& 9.6 - 0.2x + 0.5x & > 12 \\ - x - 2x & \ge -48 &&& -0.2x + 0.5x & > 12 - 9.6 \\ -3x & \ge - 48 &&& 0.3x & > 2.4 \\ x & \le {-48 \over -3} &&& x & > {2.4 \over 0.3} \\ x & \le 16 &&& x & > 8 \end{align}
\begin{align} 8 < x \le 16 & \\ \\ \text{If } x = 10, & \\ \text{No. of 50 cent coins} & = 10 \\ \text{No. of 20 cent coins} & = 48 - 10 = 38 \end{align}

(a)

\begin{align} 3x & < x - 7 \\ 3x - x & < - 7 \\ 2x & < - 7 \\ x & < {-7 \over 2} \\ x & < -3.5 \end{align}

(b)

\begin{align} 2x + 8 & > 9x \\ 2x - 9x & > - 8 \\ -7x & > - 8 \\ x & < {-8 \over -7} \\ x & < {8 \over 7} \end{align}

(c)

\begin{align} 5x & \ge 3(x - 4) \\ 5x & \ge 3x - 12 \\ 5x - 3x & \ge -12 \\ 2x & \ge -12 \\ x & \ge {-12 \over 2} \\ x & \ge -6 \end{align}

(d)

\begin{align} 4x - 2 & \le 7x - 5 \\ 4x - 7x & \le -5 + 2 \\ -3x & \le -3 \\ x & \ge {-3 \over -3} \\ x & \ge 1 \end{align}

(a)

\begin{align} x - 4 & \le 3 &&& 3x & \ge -6 \\ x & \le 3 + 4 &&& x & \ge {-6 \over 3} \\ x & \le 7 &&& x & \ge - 2 \end{align}
$$ -2 \le x \le 7 $$

(b)

\begin{align} 2x + 5 & < 15 &&& 3x - 2 & > -6 \\ 2x & < 15 - 5 &&& 3x & > -6 + 2 \\ 2x & < 10 &&& 3x & > -4 \\ x & < {10 \over 2} &&& x & > -{4 \over 3} \\ x & < 5 \end{align}
$$ -{4 \over 3} < x < 5 $$

(a)

\begin{align} 5x - 1 & < 4 &&& 3x + 5 & \ge x + 1 \\ 5x & < 4 + 1 &&& 3x - x & \ge 1 - 5 \\ 5x & < 5 &&& 2x & \ge - 4 \\ x & < {5 \over 5} &&& x & \ge {-4 \over 2} \\ x & < 1 &&& x & \ge -2 \end{align}
\begin{align} -2 & \phantom{.} \le x < 1 \\ \\ \text{Integer values of } & x \text{ are } -2, -1, 0 \end{align}

(b)

\begin{align} 2x - 5 & \ge 1 &&& 3x - 1 & < 26 \\ 2x & \ge 1 + 5 &&& 3x & < 26 + 1 \\ 2x & \ge 6 &&& 3x & < 27 \\ x & \ge {6 \over 2} &&& x & < {27 \over 3} \\ x & \ge 3 &&& x & < 9 \end{align}
\begin{align} 3 & \phantom{.} \le x < 9 \\ \\ \text{Integer values of } & x \text{ are } 3, 4, 5, 6, 7, 8 \end{align}

(a)

\begin{align} -4 & \le 2x &&& 2x & \le 3x - 2 \\ {-4 \over 2} & \le x &&& 2x - 3x & \le -2 \\ -2 & \le x &&& -x & \le -2 \\ & &&& x & \ge 2 \end{align}

$$ x \ge 2 $$

(b)

\begin{align} 1 - x & < -2 &&& -2 & \le 3 - x \\ - x & < -2 - 1 &&& x & \le 3 + 2 \\ -x & < - 3 &&& x & \le 5 \\ x & > 3 \end{align}

$$ 3 < x \le 5 $$

(c)

\begin{align} 3x - 3 & < x - 9 &&& x - 9 & < 2x \\ 3x - x & < -9 + 3 &&& x - 2x & < 9 \\ 2x & < -6 &&& -x & < 9 \\ x & < {-6 \over 2} &&& x & > - 9 \\ x & < -3 \end{align}

$$ -9 < x < -3 $$

\begin{align} \text{Let } x \text{ represent } & \text{the number of apples} \\ \\ 55 \text{ cents} & = \$ 0.55 \\ \\ \text{Total revenue} & = 0.55 \times x \\ & = 0.55x \\ \\ 0 \le \text{Profit} & \le \$ 20 \phantom{000000} [\text{'did not make a profit of more than \$ 20'}] \\ \\ 0 \le 0.55x & - 66.5 \le 20 \end{align}
\begin{align} 0 & \le 0.55x - 66.5 &&& 0.55x - 66.5 & \le 20 \\ 66.5 & \le 0.55x &&& 0.55x & \le 20 + 66.5 \\ {66.5 \over 0.55} & \le x &&& 0.55x & \le 86.5 \\ 120{10 \over 11} & \le x &&& x & \le {86.5 \over 0.55} \\ & &&& x & \le 157{3 \over 11} \end{align}
\begin{align} 120{10 \over 11} & \le x \le 157{3 \over 11} \\ \\ \therefore \text{He could have sold } & \text{between } 121 \text{ and } 157 \text{ apples (inclusive)} \end{align}

\begin{align} {1 \over 2}x - 4 & > {1 \over 3}x &&& {1 \over 6}x + 1 & < {1 \over 8}x + 3 \\ {1 \over 2}x - {1 \over 3}x & > 4 &&& {1 \over 6}x - {1 \over 8}x & < 3 - 1 \\ {1 \over 6}x & > 4 &&& {1 \over 24}x & < 2 \\ x & > 6(4) &&& x & < 24(2) \\ x & > 24 &&& x & < 48 \end{align}
\begin{align} 24 & \phantom{.} < x < 48 \\ \\ \text{Prime numbers: } & 29, 31, 37, 41, 43, 47 \end{align}

\begin{align} x + 2 & < 5 \sqrt{17} &&& 5 \sqrt{17} & < x + 3 \\ x & < 5 \sqrt{17} - 2 &&& 5 \sqrt{17} - 3 & < x \end{align}
\begin{align} 5 \sqrt{17} - 3 & < x < 5 \sqrt{17} - 2 \\ \\ 5 \sqrt{17} - 3 & \approx 17.615528 \\ 5 \sqrt{17} - 2 & \approx 18.615528 \\ \\ \text{Integer value} & \text{ of } x = 18 \end{align}

(a)

\begin{align} 3 - a & \le a - 4 &&& a - 4 & \le 9 - 2a \\ 3 + 4 & \le a + a &&& a + 2a & \le 9 + 4 \\ 7 & \le 2a &&& 3a & \le 13 \\ {7 \over 2} & \le a &&& a & \le {13 \over 3} \end{align}
$$ {7 \over 2} \le a \le {13 \over 3} $$

(b)

\begin{align} 1 - b & < b - 1 &&& b - 1 & < 11 - 2b \\ 1 + 1 & < b + b &&& b + 2b & < 11 + 1 \\ 2 & < 2b &&& 3b & < 12 \\ {2 \over 2} & < b &&& b & < {12 \over 3} \\ 1 & < b &&& b & < 4 \end{align}
$$ 1 < b < 4 $$

(c)

\begin{align} 3 - c & < 2c - 1 &&& 2c - 1 & < 5 + c \\ 3 + 1 & < 2c + c &&& 2c - c & < 5 + 1 \\ 4 & < 3c &&& c & < 6 \\ {4 \over 3} & < c \end{align}
$$ {4 \over 3} < c < 6 $$

(d)

\begin{align} 3d - 5 & < d + 1 &&& d + 1 & \le 2d + 1 \\ 3d - d & < 1 + 5 &&& d - 2d & \le 1 - 1 \\ 2d & < 6 &&& -d & \le 0 \\ d & < {6 \over 2} &&& d & \ge 0 \\ d & < 3 \end{align}
$$ 0 \le d < 3 $$

(a)

\begin{align} {a \over 4} + 3 & \le 4 &&& 4 & \le {a \over 4} + 6 \\ {a \over 4} & \le 4 - 3 &&& 4 - 6 & \le {a \over 4} \\ {a \over 4} & \le 1 &&& -2 & \le {a \over 4} \\ {a \over 4} & \le {4 \over 4} &&& {-8 \over 4} & \le {a \over 4} \\ a & \le 4 &&& -8 & \le a \end{align}
$$ -8 \le a \le 4 $$

(b)

\begin{align} {b \over 3} & \ge {b \over 2} + 1 &&& {b \over 2} + 1 & \ge b - 1 \\ {b \over 3} - {b \over 2} & \ge 1 &&& {b \over 2} - b & \ge - 1 - 1 \\ {2b \over 6} - {3b \over 6} & \ge {6 \over 6} &&& {b \over 2} - {2b \over 2} & \ge - 2 \\ {-b \over 6} & \ge {6 \over 6} &&& {-b \over 2} & \ge {-4 \over 2} \\ -b & \ge 6 &&& -b & \ge -4 \\ b & \le -6 &&& b & \le 4 \end{align}

$$ b \le - 6 $$

(c)

\begin{align} 2(1 - c) & > c - 1 &&& c - 1 & \ge {c - 2 \over 7} \\ 2 - 2c & > c - 1 &&& {7(c - 1) \over 7} & \ge {c -2 \over 7} \\ 2 + 1 & > c + 2c &&& 7(c - 1) & \ge c - 2 \\ 3 & > 3c &&& 7c - 7 & \ge c - 2 \\ {3 \over 3} & > c &&& 7c - c & \ge - 2 + 7 \\ 1 & > c &&& 6c & \ge 5 \\ 1 & > c &&& c & \ge {5 \over 6} \end{align}
$$ {5 \over 6} \le c < 1 $$

(d)

\begin{align} d - 5 & < {2d \over 5} &&& {2d \over 5} & \le {d \over 2} + {1 \over 5} \\ {5(d - 5) \over 5} & < {2d \over 5} &&& {4d \over 10} & \le {5d \over 10} + {2 \over 10} \\ 5(d - 5) & < 2d &&& {4d \over 10} & \le {5d + 2 \over 10} \\ 5d - 25 & < 2d &&& 4d & \le 5d + 2 \\ 5d - 2d & < 25 &&& 4d - 5d & \le 2 \\ 3d & < 25 &&& -d & \le 2 \\ d & < {25 \over 3} &&& d & \ge -2 \end{align}
$$ -2 \le d < {25 \over 3} $$

Weiming's payment is half the sum of the total cost and the difference between their payments. To maximise it, consider the largest permitted total cost of $210 and difference of $150, then check the share condition.

\begin{align} \text{Let } \$ x \text{ represent } & \text{the money Weiming pays} \\ \\ \text{Amount Joyce pays} & = \$ (210 - x) \end{align}
\begin{align} \text{Weiming's share} & \ge 2 \times \text{Joyce's share} &&& \text{Weiming's share} - \text{Joyce's share} & \le \$ 150 \\ x & \ge 2(210 - x) &&& x - (210 - x) & \le 150 \\ x & \ge 420 - 2x &&& x - 210 + x & \le 150 \\ x + 2x & \ge 420 &&& 2x & \le 150 + 210 \\ 3x & \ge 420 &&& 2x & \le 360 \\ x & \ge {420 \over 3} &&& x & \le {360 \over 2} \\ x & \ge 140 &&& x & \le 180 \end{align}
\begin{align} 140 & \phantom{.} \le x \le 180 \\ \\ \text{Greatest amount} & \text{ Weiming pays} = \$180 \\ \\ \text{Joyce pays} & = \$ (210 - 180) \\ & = \$ 30 \\ \\ 180 & \ge 2(30) \phantom{0000} [\text{Share condition satisfied}] \end{align}

(a) The common solution must be x < -2.

\begin{align} ax + 3 & > 2x + b &&& cx - 1 & > 5x - 4 \\ ax - 2x & > b - 3 &&& cx - 5x & > - 4 + 1 \\ (a - 2)x & > b - 3 &&& (c - 5)x & > - 3 \\ \\ \text{If } a = 1,& \phantom{.} b =5, &&& \text{If } & c = 4, \\ (1 - 2)x & > 5 - 3 &&& (4 - 5)x & > - 3 \\ - x & > 2 &&& -x & > - 3 \\ x & < - 2 &&& x & < 3 \phantom{000000} [\text{Intersection: } x < -2] \end{align}
$$ \therefore a = 1, b = 5, c = 4 $$

(b) One inequality must give x > -3 and the other must give x < 4.

\begin{align} (a - 2)x & > b - 3 &&& (c - 5)x & > - 3 \\ \\ \text{If } a = 1 &, b = -1, &&& \text{If } c & = 6, \\ (1 - 2)x & > -1 - 3 &&& (6 - 5)x & > -3 \\ -x & > - 4 &&& x & > - 3 \\ x & < 4 \end{align}
$$ \therefore a = 1, b = -1, c = 6 $$

(c)

\begin{align} (a - 2)x & > b - 3 &&& (c - 5)x & > - 3 \\ \\ \text{If } a = 3 &, b = 6, &&& \text{If } c & = 4, \\ (3 - 2)x & > 6 - 3 &&& (4 - 5)x & > -3 \\ x & > 3 &&& -x & > - 3 \\ & &&& x & < 3 \end{align}
$$ \therefore a = 3, b = 6, c = 4 $$

\begin{align} \text{Let } x \text{ represent } & \text{no. of bubble tea} \\ \\ \text{Total cost of bubble tea} & = 3.20 \times x \\ & = \$ 3.2x \\ \\ \text{No. of ice-cream cone} & = 12 - x \\ \\ \text{Total cost of ice-cream cone} & = 2.4 \times (12 - x) \\ & = 2.4(12 - x) \\ & = \$ (28.8 - 2.4x) \\ \\ 3.2 x + 28.8 - 2.4x & \le 30 \\ 0.8x & \le 30 - 28.8 \\ 0.8x & \le 1.2 \\ x & \le {1.2 \over 0.8} \\ x & \le 1.5 \\ \\ \text{Only possible to buy } & \text{0 or 1 cup of bubble tea} \\ \\ \therefore \text{She is not able to buy more} & \text{ bubble tea than ice-cream cones} \end{align}

(a)

\begin{align} 13x & < - 5x + 2 \\ 13x + 5x & < 2 \\ 18x & < 2 \\ x & < {2 \over 18} \\ x & < {1 \over 9} \end{align}

(b)

\begin{align} x - 6 & > 4x - 9 \\ -6 + 9 & > 4x - x \\ 3 & > 3x \\ {3 \over 3} & > x \\ 1 & > x \end{align}

(c)

\begin{align} 7(x - 3) & \le 2x + 1 \\ 7x - 21 & \le 2x + 1 \\ 7x - 2x & \le 1 + 21 \\ 5x & \le 22 \\ x & \le {22 \over 5} \end{align}

(d)

\begin{align} 8x + 3 & \ge 4x + 7 \\ 8x - 4x & \ge 7 - 3 \\ 4x & \ge 4 \\ x & \ge {4 \over 4} \\ x & \ge 1 \end{align}

(a)

\begin{align} 5 - a & \le a - 6 &&& a - 6 & \le 10 - 3a \\ - a - a & \le -6 - 5 &&& a + 3a & \le 10 + 6 \\ -2a & \le -11 &&& 4a & \le 16 \\ a & \ge {-11 \over -2} &&& a & \le {16 \over 4} \\ a & \ge 5.5 &&& a & \le 4 \end{align}

$$ \text{No solutions} $$

(b)

\begin{align} 4 - b & < 2b - 1 &&& 2b - 1 & < 7 + b \\ 4 + 1 & < 2b + b &&& 2b - b & < 7 + 1 \\ 5 & < 3b &&& b & < 8 \\ {5 \over 3} & < b &&& b & < 8 \end{align}
$$ {5 \over 3} < b < 8 $$

(c)

\begin{align} 4c - 1 & < {1 \over 2} &&& {1 \over 2} & \le 3c + 2 \\ 4c & < {1 \over 2} + 1 &&& {1 \over 2} - 2 & \le 3c \\ 4c & < {3 \over 2} &&& -{3 \over 2} & \le 3c \\ c & < {1 \over 4} \left(3 \over 2\right) &&& {1 \over 3}\left(-{3 \over 2}\right) & \le c \\ c & < {3 \over 8} &&& -{1 \over 2} & \le c \end{align}
$$ -{1 \over 2} \le c < {3 \over 8} $$

(d)

\begin{align} 2d + 1 & \ge d &&& d & > 3d - 20 \\ 2d - d & \ge -1 &&& 20 & > 3d - d \\ d & \ge - 1 &&& 20 & > 2d \\ & &&& {20 \over 2} & > d \\ & &&& 10 & > d \end{align}
$$ -1 \le d < 10 $$

(a)

\begin{align} 5x & > 69 - 2x &&& 27 - 2x & \ge 4 \\ 5x + 2x & > 69 &&& 27 - 4 & \ge 2x \\ 7x & > 69 &&& 23 & \ge 2x \\ x & > {69 \over 7} &&& {23 \over 2} & \ge x \\ x & > 9{6 \over 7} &&& 11{1 \over 2} & \ge x \end{align}
\begin{align} 9{6 \over 7} & \phantom{.} < x \le 11{1 \over 2} \\ \\ \text{Integer values} & \text{ of } x \text{ are 10 and 11} \end{align}

(b)

\begin{align} -10 & \le x < - 4 &&& 2 - 5x & < 35 \\ & &&& -5x & < 35 - 2 \\ & &&& -5x & < 33 \\ & &&& x & > {33 \over -5} \\ & &&& x & > -6{3 \over 5} \end{align}
\begin{align} -6{3 \over 5} & \phantom{.} < x < -4 \\ \\ \text{Integer values} & \text{ of } x \text{ are -6 and -5} \end{align}

(a)

\begin{align} 3x - 5 & < 26 &&& 26 & \le 4x - 6 \\ 3x & < 26 + 5 &&& 26 + 6 & \le 4x \\ 3x & < 31 &&& 32 & \le 4x \\ x & < {31 \over 3} &&& {32 \over 4} & \le x \\ x & < 10{1 \over 3} &&& 8 & \le x \end{align}
\begin{align} 8 & \phantom{.} \le x < 10{1 \over 3} \\ \\ \text{Integer values} & \text{ of } x \text{ are 8, 9 and 10} \end{align}

(b)

\begin{align} 3x + 2 & < 19 &&& 19 & < 5x - 4 \\ 3x & < 19 - 2 &&& 19 + 4 & < 5x \\ 3x & < 17 &&& 23 & < 5x \\ x & < {17 \over 3} &&& {23 \over 5} & < x \\ x & < 5{2 \over 3} &&& 4{3 \over 5} & < x \end{align}
\begin{align} 4{3 \over 5} & \phantom{.} < x < 5{2 \over 3} \\ \\ \text{Integer values} & \text{ of } x \text{ is 5} \end{align}

(c)

\begin{align} 3 & < 3x - 9 &&& 3x - 9 & < 4 \\ 3 + 9 & < 3x &&& 3x & < 4 + 9 \\ 12 & < 3x &&& 3x & < 13 \\ {12 \over 3} & < x &&& x & < {13 \over 3} \\ 4 & < x &&& x & < 4{1 \over 3} \end{align}
\begin{align} 4 & \phantom{.} < x < 4{1 \over 3} \\ \\ \text{No integers} & \text{ satisfy the inequality} \end{align}

(d)

\begin{align} -10 & < 7 - 2x &&& 7 - 2x & \le -1 \\ 2x & < 7 + 10 &&& 7 + 1 & \le 2x \\ 2x & < 17 &&& 8 & \le 2x \\ x & < {17 \over 2} &&& {8 \over 2} & \le x \\ x & < 8{1 \over 2} &&& 4 & \le x \end{align}
\begin{align} 4 & \phantom{.} \le x < 8{1 \over 2} \\ \\ \text{Integer values} & \text{ of } x \text{ are 4, 5, 6, 7 and 8} \end{align}

\begin{align} \text{Let } x \text{ represent } & \text{the prime number} \end{align}
\begin{align} 3(x + 5) - 7 & \le 55 &&& 4(x - 2) & > 39 \\ 3x + 15 - 7 & \le 55 &&& 4x - 8 & > 39 \\ 3x + 8 & \le 55 &&& 4x & > 39 + 8 \\ 3x & \le 55 - 8 &&& 4x & > 47 \\ 3x & \le 47 &&& x & > {47 \over 4} \\ x & \le {47 \over 3} &&& x & > 11{3 \over 4} \\ x & \le 15{2 \over 3} \end{align}
\begin{align} 11{3 \over 4} & \phantom{.} < x \le 15{2 \over 3} \\ \\ \text{The prime } & \text{number is } 13 \end{align}

Each customer who receives a goodie bag spends at least $20. For x such customers, the minimum guaranteed sales are $20x. To guarantee sales of at least $2000, require 20x to be at least 2000.

\begin{align} 45 \text{ cents} & = \$ 0.45 \\ \\ 20 \text{ cents} & = \$ 0.20 \\ \\ \text{Cost of 1 goodie bag} & = 2(0.45) + 3(0.20) \\ & = \$ 1.50 \end{align}
\begin{align} \text{Total cost of goodie bags} & \le \$200 &&& \text{Minimum guaranteed sales} & \ge \$2000 \\ (1.50)(x) & \le 200 &&& (20)(x) & \ge 2000 \\ 1.5x & \le 200 &&& 20x & \ge 2000 \\ x & \le {200 \over 1.5} &&& x & \ge {2000 \over 20} \\ x & \le 133{1 \over 3} &&& x & \ge 100 \end{align}
\begin{align} 100 & \phantom{.} \le x \le 133{1 \over 3} \\ \\ \text{Possible } & \text{value of } x \text{ is 100} \end{align}